Simple modules and their essential extensions for skew polynomial rings
 325 Downloads
Abstract
Let R be a commutative Noetherian ring and \(\alpha \) an automorphism of R. This paper addresses the question: when does the skew polynomial ring \(S = R[\theta ; \alpha ]\) satisfy the property \((\diamond )\), that for every simple Smodule V the injective hull \(E_S(V)\) of V has all its finitely generated submodules Artinian. The question is largely reduced to the special case where S is primitive, for which necessary and sufficient conditions are found, which however do not between them cover all possibilities. Nevertheless a complete characterisation is found when R is an affine algebra over a field k and \(\alpha \) is a kalgebra automorphism—in this case \((\diamond )\) holds if and only if all simple Smodules are finite dimensional over k. This leads to a discussion, involving close study of some families of examples, of when this latter condition holds for affine kalgebras \(S = R[\theta ;\alpha ]\). The paper ends with a number of open questions.
Keywords
Injective module Noetherian ring Simple module Skew polynomial ringMathematics Subject Classification
16D50 16P40 16S351 Introduction
A Noetherian ring S whose simple modules have the property that their finitely generated essential extensions are Artinian is said to satisfy property \((\diamond )\). For commutative rings S the validity of \((\diamond )\) is due to Matlis, proved in his famous 1958 paper [30]; a brief survey of work on this topic in the years since then is given below, in Sect. 1.3. This paper focusses on \((\diamond )\) for the skew polynomial rings \(S = R[\theta ; \alpha ]\), where R is a commutative Noetherian ring and \(\alpha \) is an automorphism of R, with the indeterminate \(\theta \) satisfying the relations \(\theta r = \alpha (r)\theta \) for all \(r \in R\). When such a skew polynomial ring S satisfies \((\diamond )\) turns out to be a surprisingly subtle question, which we do not completely settle here, and which leads naturally to other fundamental representationtheoretic questions concerning these rings.
We are led to pay particular attention to two separate but overlapping cases—first, when S is a primitive ring; and second, when R is an affine algebra over a field. We outline our main results for these two settings in Sects. 1.1 and 1.2 respectively. Here and throughout, given a ring R and an Rmodule V, \(E_R(V)\) will denote the Rinjective hull of V.
1.1 When S is primitive
Using relatively standard methods involving the second layer condition we show that, for every commutative Noetherian ring R, \(S = R[\theta ; \alpha ]\) satisfies \((\diamond )\) if and only if \(E_S(V)\) is locally Artinian for every simple Smodule V whose annihilator Q is induced from R. Here, “induced from R” means that \(Q = (Q \cap R)S\). This reduction is achieved in Corollary 3.5, thereby focussing attention on the case where \(S = R[\theta ; \alpha ]\) is primitive, since \(S/(Q\cap R)S \cong \overline{R}[\theta ; \overline{\alpha }]\), where \(\overline{R} = R/Q\cap R\) and \(\overline{\alpha }\) denotes the automorphism of \(\overline{R}\) induced by \(\alpha \).
Making heavy use of the characterisation of primitive skew polynomial rings [29], we prove:
Theorem 1.1
 (a)
If R has Krull dimension 0 then S satisfies \((\diamond )\).
 (b)
Suppose that R contains an uncountable field. Suppose also that either R has Krull dimension at least 2, or \(\mathrm {Spec}(R)\) is uncountable. Then S does not satisfy \((\diamond )\).
The above is Theorem 5.4; its proof occupies Sects. 4, 5. Clearly, the above necessary and sufficient conditions don’t exhaust all possibilities, and we leave the closure of the gap between (a) and (b) as one of a number of open questions raised by the paper.
1.2 When R is affine
Theorem 1.1 is however sufficient to settle the case where R is a finitely generated algebra over an uncountable field k, with \(\alpha \) a kalgebra automorphism:
Theorem 1.2
 (a)
S satisfies \((\diamond )\);
 (b)
all simple Smodules are finite dimensional kvector spaces.
This is Theorem 6.1. The direction \((b)\Rightarrow (a)\) follows from known considerations based on the second layer condition, and doesn’t require the cardinality hypothesis on k. In fact, following very welcome input from Jason Bell [2] and the application of deep modeltheoretic work [20], Theorem 6.1 can be extended to countable fields, at least when k has characteristic 0—see Remark(ii) following Theorem 6.1.
Theorem 1.2 begs the obvious question: for which commutative affine kalgebras R and kalgebra automorphisms \(\alpha \) are all the simple Smodules finitedimensional? This question seems to be hard, but we can at least answer it in the most obvious special case, namely when R is a polynomial algebra and \(\alpha \) is a linear automorphism:
Theorem 1.3
Let k be a field, t a positive integer, V a vector space over k with basis \(\{x_1, \ldots , x_t\}\) and \(\alpha \in \mathrm {GL}(t,k)\) an automorphism of V. Let \(R = k[x_1, \ldots , x_t]\), so that \(\alpha \) induces a kalgebra automorphism of R, also denoted by \(\alpha \). Then \(S := R[\theta ; \alpha ]\) satisfies \((\diamond )\) if and only if all simple Smodules are finite dimensional if and only if \(\alpha < \infty \).
Beyond this positive result, we demonstrate the complexity of the above question by means of detailed analysis of some families of examples in Sect. 7. These include an example, Example 7.4 , originally studied by Jordan [25], giving an affine domain in characteristic 0 where all the simple Smodules have finite dimension, but \(\alpha \) has infinite order. Using deep results on group algebras, we also provide many similar examples in positive characteristic (Examples 7.3). Most notably, we undertake a detailed study of the algebras \(S_{\mathbb {C}, t, \alpha } = \mathbb {C}[x_1, \ldots , x_t][\theta ; \alpha ]\) when \(t =1\) or 2 and \(\alpha \) is an arbitrary\(\mathbb {C}\)algebra automorphism. But even for \(t = 2\) our analysis is incomplete, and leads to delicate issues having connections to dynamical systems and to algebraic geometry.
1.3 Historical background
For commutative Noetherian rings \((\diamond )\) is an immediate consequence of the ArtinRees property, formally recorded as part of Matlis’s seminal 1958 paper [30] on injective modules over such rings. In 1959 Philip Hall proved \((\diamond )\) for group rings RG of finitely generated nilpotent groups G, provided R is either \(\mathbb {Z}\) or a locally finite field. In 1974 this result was extended, independently by Jategaonkar [23] and Roseblade [39], to polycyclicbyfinite groups G, building on earlier celebrated work of Roseblade [38] on the finite dimensionality of the simple RGmodules for these group rings. Hall and Roseblade were motivated by applications to the structure of finitely generated soluble groups.
Motivated by applications to Jacobson’s conjecture, Jategaonkar proved in 1974 [22] that \((\diamond )\) is satisfied by fully bounded Noetherian rings, thus incorporating Noetherian rings satisfying a polynomial identity (PI) and so generalising the commutative case. Musson [32] gave the first examples of Noetherian rings for which \((\diamond )\) fails, by showing that the group algebra kG of a polycyclicbyfinite group G over a field k which is not locally finite satisfies \((\diamond )\) only if G is abelianbyfinite, that is only if kG satisfies a PI. In so doing he thus delineated the limits of the earlier results of Hall and Roseblade.
More recent work has discussed \((\diamond )\) for differential operator rings [7, 40], downup algebras [6] and quantised Weyl algebras [8]. Musson [34] gives a brief survey of results on property \((\diamond )\) up to 2010.
1.4 Layout
Preliminary observations and notation regarding property \((\diamond )\) and skew polynomial rings are in Sect. 2. Section 3 contains a summary of the necessary background on the second layer condition, leading up to Corollary 3.5, which essentially allows us to focus on the case where \(S = R[\theta ; \alpha ]\) is a primitive ring. The analysis of primitive skew polynomial rings is contained in Sects. 4 and 5: the key result of Sect. 4 is the construction of a faithful simple Smodule whose injective hull is not locally Artinian when R is an \(\alpha \)simple domain which is not a field. This is Proposition 4.10, which then allows us to deduce Theorem 1.1 (= Theorem 5.4). In Sect. 6 Theorem 1.1 is applied in the setting where R is an affine algebra over the field k and \(\alpha \) is an algebra automorphism, to deduce Theorems 1.2 and 1.3 (= Theorems 6.1 and 6.2). Section 7 is devoted to a careful analysis of the simple modules and the prime spectra of a number of examples, and families of examples, of skew polynomial algebras over a commutative affine Noetherian domain R. These examples may well have interest beyond the immediate question at hand, namely the validity of \((\diamond )\). Finally, in Sect. 8 we gather together and briefly discuss the open questions which have arisen in the course of this work.
All rings considered are associative with identity and all modules are unitary and are right modules unless stated otherwise.
2 Preliminaries
2.1 Formulation and preservation of \((\diamond )\)
Recall that a module M is a subdirect product of a family of modules \(\{F_{\lambda }\}\) if there exists an embedding \(\iota :M\rightarrow \prod _{\lambda \in \Lambda } F_{\lambda }\) into the product of the modules \(F_{\lambda }\) such that, for each projection \(\pi _{\mu }:\prod _{\lambda \in \Lambda } F_{\lambda }\rightarrow F_{\mu }\), the composition \(\pi _{\mu }\iota \) is surjective. The following lemma is due to Hatipoğlu and Lomp, [17].
Lemma 2.1
 (a)
R satisfies \((\diamond )\);
 (b)
every right Rmodule is a subdirect product of locally Artinian modules;
 (c)
every finitely generated right Rmodule is a subdirect product of Artinian modules.
Recall that a ring extension \(T\subseteq S\) is called a finite normalizing extension if there exists a finite set \(\{s_1,\ldots , s_n\}\) of elements of S such that \(S=\sum _{i=0}^{n} s_iT\), with \(s_iT=Ts_i\), for \(i = 1, \ldots , n\). Part (a) of the next result is due to Hatipoğlu and Lomp, [17, Proposition 2.2], while (b) is adapted from the work of Hirano [19, Theorems 1.8, 1.11].
Proposition 2.2
 (a)
If T satisfies \((\diamond )\), then so does S.
 (b)
Assume also that T is Noetherian and a direct summand of S as a left Tmodule. If S satisfies \((\diamond )\), then so does T.
 (c)
If I is an ideal of S and S satisfies \((\diamond )\), then so does S / I.
Proof
 (a)
See [17, Proposition 2.2].
 (b)
Let M be a finitely generated right Tmodule, so \(M\otimes _TS\) is a finitely generated right Smodule. By hypothesis and Lemma 2.1 there exists a family \(\{M_{\lambda }\}\) of Ssubmodules of \(M\otimes _T S\) such that each \((M\otimes _T S)/M_{\lambda }\) is Artinian and \(\bigcap _{\lambda }M_{\lambda }=0\). Since T is a direct summand of S as a left Tmodule, M can be identified with a right Tsubmodule of \(M\otimes _T S\). Fix \(\lambda \). Then \(M/(M\cap M_{\lambda })\) is isomorphic to a Tsubmodule of \((M\otimes _T S)/M_{\lambda }\). By [12, Theorem 4], \((M\otimes _T S)/M_{\lambda }\) is Artinian as an Smodule if and only if it is Artinian as a Tmodule. Now, note that \(\bigcap _{\lambda }(M\cap M_{\lambda })=0\), so the result follows from Lemma 2.1(c).
 (c)
This is trivial, since \(E_{S/I}(V) \subseteq E_S(V)\) for any S / Imodule V. \(\square \)
2.2 Skew polynomial algebras of automorphism type
The following well known facts can be found in [31, \(\S \)10.6].
Lemma 2.3
 (a)
A right ideal I of R is \(\alpha \)stable if and only if \(\alpha (I)\subseteq I\).
 (b)
If I is an \(\alpha \)stable ideal of R, then IS is an ideal of S and the ring S / IS is isomorphic to \((R/I)[\theta ; {\overline{\alpha }}]\), where \(\overline{\alpha }\) is the automorphism of R / I induced by \(\alpha \). The ideal IS is prime if and only if I is \(\alpha \)prime.
 (c)
If P is a prime ideal of S such that \(\theta \notin P\) then \(P\cap R\) is an \(\alpha \)prime ideal of R.
 (d)
If P is an \(\alpha \)prime ideal of R, then there exists \(n\in \mathbb {N}\) and a minimal prime Q over P such that \( P=\bigcap _{i=0}^n \alpha ^i(Q)\).
A convenient mechanism to construct interesting simple Smodules is as follows.
Lemma 2.4
 (a)
The lattice of right \(\alpha \)ideals of R is isomorphic to the lattice of Ssubmodules of \(S/(u\theta )S\).
 (b)
The right Smodule \(S/(u\theta )S\) is simple if and only if R is \(\alpha \)simple.
 (c)
Suppose that R is also a domain with no proper idempotent ideals; for example, R could be a Noetherian domain. Then \( S/(u\theta )S\) is an Artinian right Smodule if and only if R is \(\alpha \)simple.
Proof
 (a)Notice that, for any element \(r\in R\),Let N be an \(\alpha \)stable right ideal of R. Then (1) and the centrality of u in R yield that \(J :=(u\theta )S+N\) is a right ideal of S. Let J be a right ideal of S with \((u\theta )S \subseteq J\). Let \(w\in J\). Using division by the monic polynomial \(u\theta \) we can find \(h\in S\) and \(r\in R\) such that \(w=(u\theta )h+r\) and \(J=(u\theta )S+N\) follows for \(N :=J\cap R\). Moreover, applying (1) with \(r \in N\) and using our assumptions on u, it follows that \(\alpha ^{1}(N)\subseteq N\). By Lemma 2.3(a), N is an \(\alpha \)stable right ideal of R. This completes the proof of (a).$$\begin{aligned} r\theta =\theta \alpha ^{1}(r)=(u\theta )\alpha ^{1}(r)+u\alpha ^{1}(r). \end{aligned}$$(1)
 (b)
This is a direct consequence of (a).
 (c)
Suppose that \(S/(u\theta )S\) is Artinian as a right Smodule. Let I be a proper \(\alpha \)stable ideal of R. It follows from (a) that there exists \(n\in \mathbb {N}\) such that \(I^n=I^{2n}\). The stated hypotheses force I to be (0), so R is \(\alpha \)simple. The converse is given by (b). The Krull Intersection Theorem [10, Corollary 5.4] ensures that Noetherian domains satisfy the stated hypothesis. \(\square \)
3 Reduction to the primitive case
3.1 The second layer condition
Given a nonzero module M over a right Noetherian ring T, an affiliated submodule of M is a submodule of the form \(\mathrm {Ann}_M(P)=\{m\in M:mP=0\}\), where P is an ideal of T which is maximal amongst the annihilators of nonzero submodules of M. It is easy to see that such an ideal P is a prime ideal of T, [16, Proposition 3.12]. An affiliated series for M is a series \(0=M_0 \subset M_1 \subset \cdots \subset M_n=M\) of submodules of M, such that for each \(i\in \{1,\ldots ,n\}\), \(M_i/M_{i1}\) is an affiliated submodule of \(M/M_{i1}\). The ideals \(P_i:=\mathrm {Ann}_T(M_i/M_{i1})\) are called the affiliated primes of M with respect to the given series. Full details are in [16, Chapter 8], for example.
 (a)
\(PQ\subseteq I\subset P\cap Q\), and \((P\cap Q)/I\) is torsionfree as a left T / Pmodule and as a right T / Qmodule; or
 (b)
\(P\subset Q\) and \(MP=0\).
A Noetherian ring T is said to satisfy the (right) strong second layer condition (s.s.l.c.) if, for every prime ideal Q of T, only case (a) can occur in the setting of (2). The formally weaker (right) second layer condition (s.l.c.) holds for T if, for all primes Q, only case (a) occurs when U is in addition required to be T / Qtorsionfree.
Those Noetherian rings satisfying the s.s.l.c. form an important and large subclass. For our purposes, the key result in this direction is the following proposition. Recall (for example, from [16, page 224]) that a Noetherian ring T is ARseparated if, for every prime ideal Q of T and ideal I with \(Q \subset I \subset T\), there is an ideal J of T with \(Q \subset J \subseteq I\) such that J / Q has the ArtinRees property in T / Q.
Proposition 3.1
[16, Theorem 13.4] If the Noetherian ring T is ARseparated, then it satisfies the s.s.l.c.
For example, Noetherian rings satisfying a polynomial identity and enveloping algebras of finite dimensional solvable Lie algebras are ARseparated, [24]. Of more relevance for us, however, is:
Proposition 3.2
 (a)
Then S is ARseparated, and hence satisfies s.s.l.c.
 (b)
Let P be a prime ideal of S such that \((P\cap R)S=P\). Then P has the ArtinRees property. In particular, if \(P\rightsquigarrow Q\) or \(Q\rightsquigarrow P \), then \(Q = P\).
3.2 The second layer condition and \((\diamond )\)
Lenagan’s Lemma, [16, Theorem 7.11], guarantees that if T is a Noetherian ring and P and Q are prime ideals of T with \(P\rightsquigarrow Q\), then T / P is Artinian if and only if T / Q is Artinian. From this and Jategaonkar’s Main Lemma the following wellknown consequence follows easily:
Proposition 3.3
Let T be a Noetherian ring satisfying the s.l.c. and let V be a simple right Tmodule with \(Q := \mathrm {Ann}_T(V)\). Suppose that T / Q is Artinian. Then every finitely generated essential extension of V is Artinian.
We are now in a position to deal with property \((\diamond )\) for many simple modules over skew polynomial rings.
Theorem 3.4
Let R be a commutative Noetherian ring, \(\alpha \in Aut(R)\) and \(S=R[\theta ;\alpha ]\). Let V be a simple right Smodule and let \(Q=ann_S(V)\). If \((Q\cap R)S\subset Q\), then S / Q is Artinian and (hence) every finitely generated essential extension of V is Artinian.
Proof
Suppose first that \(\theta \in Q\). Then \(Q/\theta S\) is a primitive ideal of the commutative ring \(S/\theta S\simeq R\), so S / Q is a field. Therefore the desired property of V follows from Propositions 3.2(a) and 3.3.
Suppose now that \(\theta \notin Q\) and \((Q\cap R)S\subset Q\). Then \(Q\cap R\) is \(\alpha \)prime by Lemma 2.3(c). By [21, Theorem 4.3] the order of the automorphism of \(R/(Q\cap R)\) induced by \(\alpha \) is finite. Hence, by [9, Corollary 10], \(S/(Q\cap R)S\simeq (R/(Q\cap R))[\theta ; \alpha _{R/(Q\cap R)}]\) is a ring satisfying a polynomial identity. By Kaplansky’s Theorem [4, Theorem I.13.3] applied to the primitive ideal \(Q/(Q\cap R)S\) of \(S/(Q\cap R)S\), we deduce once again that S / Q is Artinian. Thus, again, \(E_S(V)\) is locally Artinian by Propositions 3.2(a) and 3.3. \(\square \)
The following corollary of Theorem 3.4 in large part reduces the analysis of \((\diamond )\), for skew polynomial rings S of automorphism type, to the case where S is a primitive ring. Recall that a Noetherian ring T is polynormal if for every pair of distinct ideals I and J of T with \(I\subset J\), there is an element \(a\in J{\setminus }I\) such that \(aS + I= Sa + I\).
Corollary 3.5
 (a)
S satisfies \((\diamond )\);
 (b)
\(E_S(V)\) is locally Artinian for every simple right Smodule V whose annihilator Q satisfies \(Q=(Q\cap R)S\);
 (c)
every primitive factor of S of the form \(S/(P\cap R)S\) satisfies \((\diamond )\).
Proof
(a)\(\Leftrightarrow \)(b) is a consequence of Theorem 3.4. That (a)\(\Rightarrow \)(c) follows from Proposition 2.2(c).
If S satisfies one of the additional hypotheses then (c)\(\Rightarrow \)(a) follows from Theorem 3.4 combined with [24, Theorem 9.3.4]; see also [5, Lemmas 6.1 and 6.3]. \(\square \)
4 The primitive case
4.1 Primitive skew polynomial rings
We begin by recalling the results of [29], where Leroy and Matczuk presented necessary and sufficient conditions for the primitivity of \(S =R[\theta ; \alpha ]\); see also [25].
Definition 4.1
 (a)
For all \(n\ge 1\), \(N_n^{\alpha }(a):=a\alpha (a)\ldots \alpha ^{n1}(a)\ne 0\).
 (b)
For every nonzero \(\alpha \)stable ideal I of T, there exists \(n\ge 1\) such that \(N_n^{\alpha }(a)\in I\).
From the definition it follows easily that an \(\alpha \)special ring is \(\alpha \)prime. Clearly, an \(\alpha \)simple ring is \(\alpha \)special, with 1 as \(\alpha \)special element in this case. However, there are \(\alpha \)special rings which are not \(\alpha \)simple. Consider, for example, the ring R of formal power series k[[X]], where k is a field containing an element q which is not a root of unity. Let \(\alpha \) be the kalgebra automorphism defined by setting \(\alpha (X) := qX\). Then X is an \(\alpha \)special element of R, but \(\langle X \rangle \) is a proper \(\alpha \)ideal of R.
Here is the characterisation of primitivity:
Theorem 4.2
[29, Theorem 3.10] Let R be a commutative Noetherian ring and let \(\alpha \in \mathrm {Aut}(R)\). Then \(S=R[\theta ; \alpha ]\) is primitive if and only if R is \(\alpha \)special and \(\alpha \) has infinite order.
In the proof of Theorem 4.2, given a Noetherian PI ring R and an automorphism \(\alpha \) of R with \(\alpha \)special element a, the authors build a simple faithful module over \(S=R[\theta ; \alpha ]\) of the form S / M, where M is a maximal right ideal of S containing \((1a\theta )S\). Following similar ideas, we show in the proposition below that, when R is commutative, \((1a\theta )S\) is actually a maximal right ideal; this will be important for us in the sequel. Note also that the proposition provides a proof that the conditions on \(\alpha \) in Theorem 4.2 are sufficient for primitivity of S when R is commutative.
Proposition 4.3
Let R be a commutative Noetherian ring and let \(\alpha \in Aut(R)\) be such that R is \(\alpha \)special with a an \(\alpha \)special element of R. Let \(S=R[\theta ;\alpha ]\). Then \((1a\theta )S\) is a maximal right ideal of S. If in addition \(\alpha \) has infinite order, then \(V:=S/(1a\theta )S\) is a faithful right Smodule.
Proof
To prove the last statement, assume that \(\alpha \) is of infinite order. Let \(P=\mathrm {Ann}_S(V)\). Then \(P\subseteq (1a\theta )S\), so \(P\cap R= (0)\) and P does not contain \(\theta \). By [21, Theorem 4.3] as R is \(\alpha \)prime and \(\alpha \) has infinite order, every nonzero prime ideal of S which intersects R in (0) contains \(\theta \), so \(P=(0)\). \(\square \)
Remark 4.4
4.2 Reduction to the case where R is a domain
To achieve the reduction as in the title of the subsection we need the following lemma:
Lemma 4.5
 (a)
R is \(\alpha \)prime, and so there exist \(n \in \mathbb {N}\) and a prime ideal Q of R such that \(\{Q,\alpha (Q), \ldots , \alpha ^{n1}(Q)\}\) is the set of minimal primes of R, with \(\cap _{i=0}^{n1}\alpha ^i(Q) = (0)\).
 (b)
In the setting of (a), if S is primitive then \((R/Q)[\theta ^n; \alpha ^n]\) is primitive.
Proof
 (a)
This is clear from Lemma 2.3(c),(d).
 (b)With the notation of (a), let \(T := R[\theta ^n ; \alpha ^n]\). Notice that QT and its \(\alpha \)conjugates are the minimal prime ideals of T. Suppose that V is a faithful simple right Smodule. By [12, Theorem 4] V has a (finite) composition series as a Tmodule,Set \(P_i := \mathrm {Ann}_T(V_i/V_{i1})\), for \(i = 1, \ldots , t\), so that$$\begin{aligned} 0 = V_0 \subset V_1 \subset \cdots \subset V_t = V.\end{aligned}$$Since V is by hypothesis a faithful Smodule, \( P_tP_{t1}\ldots P_1 = (0)\). Hence there exists j, \(1 \le j \le t\), with \(P_j \subseteq QT\), and the minimality of the prime ideal QT of T ensures that \(P_j = QT\). That is, \(V_j/V_{j1}\) is a faithful simple T / QTmodule, as required. \(\square \)$$\begin{aligned} V(P_t P_{t1} \ldots P_1) = 0. \end{aligned}$$
The above lemma, with an equivalence in (b), can be found as [29, Corollary 2.2] with a different proof.
Lemma 4.6
 (a)
Let \(n \ge 1\) and let \(T=R[\theta ^n; \alpha ^n]\). Then S satisfies \((\diamond )\) if and only if T satisfies \((\diamond )\) .
 (b)
Suppose that R is \(\alpha \)prime, with Q and n chosen as in Lemma 4.5(a). If S satisfies \((\diamond )\) so does \(T/QT\simeq (R/Q)[\theta ^n; \alpha ^n]\).
4.3 \((\diamond )\) When R is an \(\alpha \)simple domain
To investigate which primitive skew polynomial rings over a commutative Noetherian domain R satisfy \((\diamond )\), we first consider the case when R is \(\alpha \)simple. We preface the proposition with three lemmas needed for its proof.
Lemma 4.7
Proof
For each i, there is a positive integer \(n_i\) such that \(\{\alpha ^{n_it}(P_i) : t \in \mathbb {Z}\} \cap \mathcal {P} = \{P_i\}\). Thus \(n := \prod _i n_i\) is as required. \(\square \)
Lemma 4.8
Let \(\alpha \) be an automorphism of the commutative ring R, let \(S = R[\theta ; \alpha ]\), and let \(\rho \) be a nonzero nonunit of R. Then \(S/\rho S\) is not an Artinian right Smodule.
Proof
Lemma 4.9
Proof
Since \(S = R \oplus (1  \theta )S\) as right Rmodules and \(S \cong \rho S\) as right Smodules, the desired expression using coset representatives \(\{ \rho b : b \in R \}\) for the elements of the submodule \(\rho S/\rho (1  \theta )S\) of M is clear. Representation of elements of M by the listed elements follows, since \(S/\rho S \cong R/\rho R \otimes _R S\) as right Smodules. \(\square \)
Proposition 4.10
Let R be a commutative Noetherian domain which is not a field, \(\alpha \) an automorphism of R and suppose that R is \(\alpha \)simple. Then \(S= R[\theta ;\alpha ]\) does not satisfy \((\diamond )\). Moreover, there is \(n\in \mathbb {N}\) such that \(S/(1\theta ^n)S\) is a finite length Smodule whose injective hull is not locally Artinian.
Proof
Let \(\rho \in R\) be a nonzero nonunit of R, and let \(\mathcal {P} = \{P_1, \ldots , P_t \}\) be the set of annihilator primes in R of the Rmodule \(R/\rho R\). Thus \(\rho R \subseteq P_i\) for all \(i = 1, \ldots , t\), and the inverse image in R of every minimal prime of \(R/\rho R\) is in \(\mathcal {P}\), but there may also be further primes in \(\mathcal {P}\). We divide the proof in two cases.
Sublemma 1
If \(\ell (p) > 1,\) then pS contains a nonzero element \(\widehat{p}\) with \(0< \ell (\widehat{p}) < \ell (p)\).
Proof of Sublemma 1
Sublemma 2
Let \(p \in M\) be written in normal form (6), with \(\ell (p) = 1\)  that is, \(\ell = k\) and \(r_{\lambda _k} \ne 0.\) Then there exists \(u \in R\) such that \(0 \ne pu \in V.\)
Proof of Sublemma 2
Thus (5) follows at once from the two sublemmas, and \(V=S/(1\theta )S\) is a simple Smodule with an injective hull that is not locally Artinian.
Case 2. Assume that (4) does not hold. Note that if R is \(\alpha \)simple, then R is also \(\alpha ^m\)simple for any \(m\in \mathbb {N}\). Indeed if I is \(\alpha ^m\)stable proper ideal of R, then \(I\alpha (I)\ldots \alpha ^{m1}(I)=0\) as it is an \(\alpha \)stable proper ideal, and so \(I = (0)\) since R is a domain. Hence also every nonzero prime ideal has an infinite \(\alpha ^m\)orbit. Choose \(n\in \mathbb {N}\) such that the conclusion of Lemma 4.7 holds for \(\alpha \) and \(\mathcal {P}\). By the above we can apply Case 1 to \(S'=R[\theta ^n; \alpha ^n]\) and obtain that \(V=S'/(1\theta ^n)S'\) is a simple \(S'\)module with an injective hull that is not locally Artinian.
\(\square \)
In fact it can be shown, using an argument based on [44, Exercise 31 on page 112] that \(E_{S'}(V)\otimes _{S'} S=E_S(V\otimes S)\) but we do not give details here since we do not need that fact.
5 Necessary and sufficient conditions for \((\diamond )\)
In handling primitive skew polynomial rings our approach is to strengthen the \(\alpha \)special property, guaranteed by Theorem 4.2 for such a primitive ring, to the stronger \(\alpha \)simple condition. This can be achieved by localising at the smallest \(\alpha \)stable Ore set \(\mathcal {A}\) of S containing the \(\alpha \)special element. However, to apply Proposition 4.10 after this localisation, we need to exclude the possibility that \(R\mathcal {A}^{1}\) is a field. To achieve this, we need to assume that R is “sufficiently big” in terms both of the height and the “width” of its lattice of prime ideals, and that it contains an uncountable field. The technicalities for this are provided by the lemmas of Sect. 5.1. The key results are then Proposition 5.3 and Theorem 5.4.
5.1 Localisation lemmas
Lemma 5.1
Let T be a ring and \({\mathcal A}\) a multiplicatively closed Ore subset of regular elements of T. If there exists a right Tmodule V such that \(E_{T{\mathcal A}^{1}}(V\otimes _T T{\mathcal A}^{1})\) is not locally Artinian as a \(T{\mathcal A}^{1}\)module, then \(E_T(V/\tau (V))\) is not locally Artinian, where \(\tau (V)\) denotes the \(\mathcal A\)torsion submodule of V.
Proof
Suppose that \(E_{T{\mathcal A}^{1}}(V\otimes T{\mathcal A}^{1})\) is not locally Artinian as a \(T{\mathcal A}^{1}\)module. In particular V is not \({\mathcal A}\)torsion. Let \(\tau (V)\) be the \({\mathcal A}\)torsion submodule of V, then \(\frac{V}{\tau (V)}\otimes T{\mathcal A}^{1}\simeq V\otimes T{\mathcal A}^{1}\). Thus replacing V by \(V/\tau (V)\) we may assume that \(\tau (V)=0\).
The example (already featuring in Sect. 4) of \(R = k[[X]]\) and \(\mathcal {A} = \{X^i : i \ge 0 \}\) should be borne in mind in conjunction with the next lemma.
Lemma 5.2
Let R be a commutative Noetherian domain which is an algebra over an uncountable field, but is not itself a field. Suppose that there exists a countable multiplicatively closed subset \({\mathcal A}\) of \(R{\setminus }\{0\}\) such that \(R{\mathcal A}^{1}\) is the quotient field of R. Then R has Krull dimension 1 and \(\mathrm {Spec}(R)\) is countable.
Proof
Assume R and \({\mathcal A}\) are as above. By [41, Proposition 2.5] every Noetherian ring containing an uncountable field has the countable prime avoidance property. It follows from [27, Theorem 3.8] that \(\mathrm {Spec}(R)\) is countable and that each nonzero prime ideal is maximal. \(\square \)
5.2 \((\diamond )\) when S is primitive
Recall that the multiplicatively closed \(\alpha \)stable Ore subset \({\mathcal A}\) of S generated by an \(\alpha \)special element a of R was defined in Remark 4.4.
Proposition 5.3
Let R be a commutative Noetherian domain, \(\alpha \) an automorphism of R. Suppose that \(S=R[\theta ; \alpha ]\) is primitive, with \(\alpha \)special element a and associated Ore subset \({\mathcal A}\). Suppose that \(R{\mathcal A}^{1}\) is not a field. Then S does not satisfy (\(\diamond \)).
Proof
Set \(S'=R[\theta ^n; \alpha ^n]\). Since for any \(\alpha ^n\)stable ideal I of R, the ideal \(I\alpha (I)\ldots \alpha ^{n1}(I)\) is \(\alpha \)stable, it is clear that R is \(\alpha ^n\)special with the special element \(b=a\alpha (a)\alpha ^2(a)\ldots \alpha ^{n1}(a)\). Thus \(S'\) is primitive and by Proposition 4.3, \(V=S'/(1b\theta ^n)S'=S'/(1(a\theta )^n)S'\) is simple as a \(S'\)module since \(b\theta ^n=(a\theta )^n\).
By similar arguments to the ones at the end of the proof of Proposition 4.10, \(W:=V\otimes S=S/(1(a\theta )^n)S\) is of finite length as Smodule. By Lemma 5.1 the injective hull of the finite length module \(W/\tau (W)\) is not locally Artinian and the result follows. \(\square \)
We can now give necessary and sufficient conditions for \(S = R[\theta ;\alpha ]\) to satisfy \((\diamond )\) when S is primitive. Unfortunately, however, these two conditions do not together cover all possibilities.
Theorem 5.4
 (a)
If R has Krull dimension 0 then S satisfies \((\diamond )\).
 (b)
Suppose that R contains an uncountable field. Suppose also that either R has Krull dimension at least 2, or \(\mathrm {Spec}(R)\) is uncountable. Then S does not satisfy \((\diamond )\).
Proof
 (a)
When R has Krull dimension 0, S has Krull dimension 1 by [16, Theorem 15.19]. Since S is a prime Noetherian ring of Krull dimension 1, \((\diamond )\) follows easily (see for instance [32, Proposition 5.5]).
 (b)
Suppose that R contains an uncountable field, and that R has Krull dimension 2 or more, or has uncountably many prime ideals. Let Q be a minimal prime ideal of R, let \(n \in \mathbb {N}\) be such that the minimal primes of R are \(\{\alpha ^i (Q) : 0 \le i \le n1\}\), and set \(T := R[\theta ^n ; \alpha ^n]\). By Lemma 4.5(b) T / QT is also primitive, and by Lemma 4.6(b) it is enough to show that T / QT does not satisfy \((\diamond )\). In other words, in proving (b), we can pass to \((R/Q)[\theta ^n; \alpha ^n]\), observing that R / Q will like R contain an uncountable field, and have Krull dimension at least 2 or have uncountably many prime ideals, just as R does. That is, we may assume additionally that R is a domain in proving (b). Let a be the \(\alpha \)special element of R guaranteed by Theorem 4.2, with \(\mathcal {A}\) the \(\alpha \)stable Ore set it generates. By Lemma 5.2\(R\mathcal {A}^{1}\) cannot be the quotient field of R. Proposition 5.3 now implies that S does not satisfy \((\diamond )\). \(\square \)
It is straightforward to show that in Theorem 5.4 under the hypothesis (a), the ring R is a finite direct sum of isomorphic fields.
6 Property \((\diamond )\) when R is affine
The gap between parts (a) and (b) of Theorem 5.4 can be closed so as to determine completely the occurrence of \((\diamond )\) for \(S = R[\theta ; \alpha ]\), provided R contains an uncountable field and has no \(\alpha \)invariant factors of Krull dimension 1 whose spectrum is countable. In particular, this allows us to completely settle the matter when R is an affine algebra over an uncountable field, as follows.
Theorem 6.1
 (a)
S satisfies \((\diamond )\);
 (b)
all simple Smodules are finite dimensional kvector spaces.
Proof
(b)\(\Rightarrow \)(a) This follows immediately from Propositions 3.2 and 3.3 and does not require that k is uncountable.
(a)\(\Rightarrow \)(b) Suppose that S satisfies \((\diamond )\). Let V be a simple Smodule and \(P=\mathrm {Ann}_S(V)\).
If \(\theta \in P\), then \(P/\theta S\) is a primitive ideal of the commutative affine kalgebra R, so V is finite dimensional over k, thanks to the Nullstellensatz.
Suppose next that \((P\cap R)S\subset P\) and \(\theta \notin P\). Then [21, Theorem 4.3] implies that \(S/(P\cap R)S\) satisfies a polynomial identity. Since \(S/(P\cap R)S\) is also by hypothesis and construction a Noetherian affine kalgebra, V is finite dimensional over k by Kaplansky’s theorem, [4, Theorem I.13.3].
Suppose finally that \((P\cap R)S=P\) and let Q as usual denote a minimal prime over \(P\cap R\) in R. If \(R/P \cap R\) has Krull dimension 0, it is a finite direct sum of copies of the field \(R/Q := F\), by Lemma 2.3(d). Hilbert’s Nullstellensatz ensures that F is finite dimensional over k, so that some finite power, say n, of \(\alpha \) not only fixes Q, but then also induces the identity on F. Hence, S / P is a finite (free) module over the commutative ring \((R/P \cap R)[\theta ^n]\), so S / P satisfies a polynomial identity and Kaplansky’s theorem applies. Suppose on the other hand that \(R/P \cap R\) has Krull dimension at least 1. Then, for example using Lying Over and the Noether normalisation theorem [10, Proposition 4.15 and \(\S \)8.2.1, Theorem A1], \(\mathrm {Spec}(R/Q)\) is uncountable since k is uncountable. By Theorem 5.4(b), this contradicts property \((\diamond )\) for S. So no such primitive ideals P can exist in S and (b) follows. \(\square \)
Remark
 (i)Notice that \((b)\Rightarrow (a)\) of this theorem is valid, with the same proof, for an arbitrary commutative Noetherian kalgebra over any field. Indeed, if we change statement (b) tothen a small adjustment to the argument confirms that \((b')\Rightarrow (a)\) is true for all commutative Noetherian coefficient rings R.$$\begin{aligned} (b') \quad S/\mathrm {Ann}_S(V) \textit{ is Artinian for all simple } S\textit{modules } V, \end{aligned}$$
 (ii)
Following circulation of an earlier draft of this paper, Jason Bell [2] has explained to us how to remove the hypothesis that k is uncountable from Theorem 6.1, at least when k has characteristic 0. The key point is to obtain a replacement for Proposition 5.3 in the case where R is kaffine and k has characteristic 0. The substitute result is:
Proposition 5.3’Let k be a field of characteristic 0 and let R be a commutative kaffine domain which is not a field. Let \(\alpha \) be a kalgebra automorphism of R. Suppose that \(S=R[\theta ; \alpha ]\) is primitive, with \(\alpha \)special element a and associated Ore subset \({\mathcal A}\). Then \(R{\mathcal A}^{1}\) is not a field and hence S does not satisfy (\(\diamond \)).
With Proposition 5.3’ to hand, one can follow the proof of Theorem 6.1 (a)\(\Rightarrow \)(b), invoking 5.3’ to handle the crucial case where \(P = (P \cap R)S\) with Q a minimal prime over \(P \cap R\), by deducing that then R / Q is a field.
Here is a brief indication of how to prove Proposition 5.3’. First observe that the automorphism \(\alpha \) and the \(\alpha \)special element a are defined in a finitely generated field extension K of \(\mathbb {Q}\). We can then work over a finitely generated \(\mathbb {Z}\)algebra T whose field of fractions is K. One can specialise modulo the maximal ideals of T, so to \(\mathbb {F}_q\)algebras for prime powers q, and apply modeltheoretic work of Hrushovski, [20, Corollary 1.2], using the \(\mathbb {F}_q\)Frobenius to see that that the \(\alpha \)special configuration with the localisation at the image of \(\mathcal {A}\) being a field is impossible. This conclusion then lifts back to R, proving 5.3’.
Let k be any field, let R be a commutative affine kalgebra and let \(\alpha \) be a kalgebra automorphism of R. In the light of Theorem 6.1 it is natural to ask exactly what conditions on R and \(\alpha \) are required for (b) of Theorem 6.1 to hold. This appears to be quite a subtle question, which we discuss further in Sect. 7. First, we give here an important special case in which a complete answer is available:
Proposition 6.2
Let k be a field, t a positive integer, V a vector space over k with basis \(\{x_1, \ldots , x_t\}\) and \(\alpha \in \mathrm {GL}(t,k)\) an automorphism of V. Let \(R = k[x_1, \ldots , x_t]\), so that \(\alpha \) induces a kalgebra automorphism of R, also denoted by \(\alpha \). Then \(S := R[\theta ; \alpha ]\) satisfies \((\diamond )\) if and only if the order \(\alpha \) of the automorphism \(\alpha \) is finite.
Proof
Suppose \(\alpha  = n < \infty \). Since the commutative Noetherian ring \(R[\theta ^n]\) satisfies \((\diamond )\), Lemma 4.6(a) implies that S satisfies \((\diamond )\).
Notice also that this completes the proof of the proposition in the case where k, or equivalently L, has positive characteristic, since then the matrix of \(\alpha \) lies in GL(t, E) for some finite field E.
7 Examples and discussion
 \((\bullet )\)

S satisfies \((\diamond )\).
 \((\bullet )\)

Every simple Smodule has finite dimension over k.
 \((\bullet )\)

S satisfies a polynomial identity.
 \((\bullet )\)

\(\alpha \) has finite order.
We first consider, in Sects. 7.1 and 7.2, the application of Theorem 6.1 to the algebras \(S_{k,n,\alpha }:= k[x_1, \ldots , x_n][\theta ; \alpha ]\), where k is a field, n is a positive integer, and \(\alpha \) is as a kalgebra automorphism of \(k[x_1, \ldots , x_n]\). Note that if \(\xi \in A=Aut(k[x_1, \ldots , x_n])\), the group of kalgebra automorphisms of \(R=k[x_1, \ldots , x_n]\), then \(\xi \) extends to an isomorphism from \(R[\theta ;\alpha ]\) to \(R[\theta ; \xi \alpha \xi ^{1}]\). Thus, when convenient, we may replace \(\alpha \) by its conjugate. Here, we will only consider \(n=1\) and \(n=2\), but even for these small values of n the situation turns out to be surprisingly delicate. The representation theory of \(S_{k,n,\alpha }\) likely has close interactions with the dynamical properties of \(\alpha \), as studied for example in [11] and subsequent works; see for instance [37]. Even for the case \(n=2\) we are unable to determine whether the first bullet point above is equivalent to the third and fourth.
We then turn in Sect. 7.3 to algebras \(S=R[\theta ;\alpha ]\) occurring as subalgebras of group algebras kG of torsionfree polycyclic groups G, where k is algebraic over a field of p elements, p prime. Here, thanks to deep results on the representation theory of these group algebras, we can easily construct many examples where the first two of the above bullet points hold, but the second two do not.
In the final subsection, Sect. 7.4, we describe an example due to Jordan (cf. [25]), which shows that there is no equivalence of the four bullet points for affine algebras in characteristic 0.
7.1 \((\diamond )\) for \(S_{k,1,\alpha }\)
Here k can be an arbitrary field. As is easy to confirm, the group A of kalgebra automorphisms of k[x] consists of the affine automorphisms, mapping x to \(\beta x + \gamma \), for \(\beta ,\gamma \in k\) with \(\beta \) nonzero.
Proposition 7.1
 (a)
\(S_{k,1,\alpha }\) satisfies \((\diamond )\);
 (b)
every simple \(S_{k,1,\alpha }\)module is finite dimensional over k;
 (c)
\(S_{k,1,\alpha }\) is a finite module over its centre;
 (d)
\(\alpha \) is of finite order, i.e if \(\beta \ne 1\) then \(\beta \) is a root of unity and if \(\beta =1\), \(\gamma =0\) when k has characteristic 0.
Proof
The implications (d)\(\Rightarrow \)(c)\(\Rightarrow \)(b)\(\Rightarrow \)(a) are standard. Indeed suppose (d) holds. Then \(\alpha \) is of finite order and by Noether’s Theorem [42, Theorem 2.3.1], (c) follows easily. Moreover S satisfies a polynomial identity and by Kaplansky’s Theorem [4, Theorem I.13.3] we have that (c)\(\Rightarrow \)(b). The implication (b)\(\Rightarrow \)(a) follows as in Theorem 6.1.
(a)\(\Rightarrow \)(d) Suppose that \(S_{k,1,\alpha }\) satisfies \((\diamond )\). Note that if \(\beta \ne 1\) then \(\alpha \) is conjugate to \(\alpha '\in Aut(k[x])\) such that \(\alpha '(x)=\beta x\). Thus by Proposition 6.2 (or [8, Theorem 3.1]) it follows that \(\beta \) is a root of unity.
If \(\beta =1\) and k has characteristic 0 then, by Proposition 4.10, \(\gamma =0\). \(\square \)
7.2 \((\diamond )\) for \(S_{\mathbb {C},2,\alpha }\)
 (i)
\(x\mapsto \lambda x; \quad y\mapsto \mu y, \quad \lambda , \mu \in \mathbb {C}{\setminus }\{0\}\);
 (ii)
\(x\mapsto \lambda x; \quad y\mapsto y+c,\quad \lambda ,c\in \mathbb {C}{\setminus }\{0\} \);
 (iii)
\(x\mapsto \lambda x+ \sum _{\{i:\lambda =\mu ^i\}} \eta _i y^i; \quad y\mapsto \mu y,\quad \eta _i\in {\mathbb {C}}, \, \lambda , \mu \in \mathbb {C}{\setminus }\{0\}\).
Lemma 7.2
 (a)
\(S_{\mathbb {C},2,\alpha }\) satisfies \((\diamond )\);
 (b)
\(\alpha \) is conjugate to an automorphism of type (i), with \(\lambda \) and \(\mu \) both roots of unity;
 (c)
\(\alpha \) has finite order;
 (d)
\(S_{\mathbb {C},2,\alpha }\) is a finite module over its centre;
 (e)
every simple \(S_{\mathbb {C},2,\alpha }\)module is a finite dimensional kvector space.
Proof
The implication (b)\(\Rightarrow \)(c) is clear and (c)\(\Rightarrow \)(d)\(\Rightarrow \)(e) follow as in the first paragraph of the proof of Proposition 7.1. Finally, (e)\(\Rightarrow \)(a) is a special case of (b)\(\Rightarrow \)(a) of Theorem 6.1.
To prove (a)\(\Rightarrow \)(b) suppose that \(S_{\mathbb {C},2,\alpha }\) satisfies \((\diamond )\). If \(\alpha \) is of type (ii), then \(xS_{\mathbb {C},2,\alpha }\) is an ideal of \(S_{\mathbb {C},2,\alpha }\) and \(S_{\mathbb {C},2,\alpha }/xS_{\mathbb {C},2,\alpha }\) is isomorphic to \({\mathbb C}[y][\theta ; \beta ]\) where \(\beta (y)=y+c\) for \(c\ne 0\). By Propositions 7.1 and 2.2(c), \(S_{\mathbb {C},2,\alpha }\) does not satisfy \((\diamond )\), contradicting our hypothesis.
Suppose alternatively that \(\alpha \) is of type (iii) but not type (i). Since \(yS_{\mathbb {C},2,\alpha }\) is an ideal of \(S_{\mathbb {C},2,\alpha }\) and \(S_{\mathbb {C},2,\alpha }/yS_{\mathbb {C},2,\alpha }\) is isomorphic to a quantum plane with parameter \(\lambda \). By Propositions 7.1 and 2.2(c), \(\lambda \) is a root of unity. Since \(S_{\mathbb {C},2,\alpha }\) is by hypothesis not type (i), this forces \(\mu \) also to be a root of unity. By Proposition 2.2 the subalgebra \(\mathbb {C}[x,y][\theta ^n ; \alpha ^n]\) of \(S_{\mathbb {C},2,\alpha }\) also satisfies \((\diamond )\), so we may pass to that subalgebra for a suitable choice of n, and thus assume that \(\mu =1=\lambda \), (but keep the notation \(S_{\mathbb {C},2,\alpha } = \mathbb {C}[x,y][\theta ; \alpha ]\)). Let \(\tau \in \mathbb {C}\) be a root of the polynomial \(\sum _{\lambda =\mu ^i} \eta _i y^i1\). Then \((y  \tau )S_{\mathbb {C},2,\alpha }\) is an ideal of \(S_{\mathbb {C},2,\alpha }\), with \(S_{\mathbb {C},2,\alpha }/(y\tau )S_{\mathbb {C},2,\alpha }\) isomorphic to \({\mathbb C}[x][\theta ;\overline{\alpha }]\) and \(\overline{\alpha }(x)=x+1\). A contradiction once again follows from Propositions 7.1 and 2.2(c). \(\square \)
Let us examine first the prime spectrum of \(S_{\mathbb {C},2,\alpha }\) when \(\alpha \) is square. We follow initially in the proof of the following lemma the argument of [25, proof of Proposition 7.8, first paragraph].
Lemma 7.3
 (a)
\(\mathcal {V}(\theta )\) is homeomorphic to \(\mathrm {Spec}(\mathbb {C}[x,y])\).
 (b)
\(\mathcal {C}(\theta )\) is homeomorphic to \(\mathrm {Spec}(T_{\mathbb {C},2,\alpha })\), where \(T_{\mathbb {C},2,\alpha }:= \mathbb {C}[x,y][\theta ^{\pm 1}; \alpha ]\).
 (c)\(\mathrm {Spec}(T_{\mathbb {C},2,\alpha })\) is partitioned into the following three disjoint subsets.
 (i)
An uncountable set of coArtinian maximal ideals \(\{M_{j,\lambda }: j \in \mathbb {N}, \lambda \in \mathbb {C}\}\), all of height 2;
 (ii)A countably infinite set of height one mutually comaximal prime ideals \(\{P_j : j \in \mathbb {N} \}\), with$$\begin{aligned} \bigcap _{j \in \mathbb {N}}P_j \quad = \quad (0); \end{aligned}$$
 (iii)
(0).
 (i)
 (d)For each \(j \in \mathbb {N}\), there exists a positive integer \(n_j\) and a finite \(\langle \alpha \rangle \)orbit \(\{Q_{j,0}, Q_{j,1} = \alpha (Q_{j,0}),\ldots , Q_{j, n_j  1} = \alpha ^{n_j1}(Q_{j,0})\}\) of maximal ideals of \(\mathbb {C}[x,y]\), such that$$\begin{aligned} P_j \quad = \quad \left( \bigcap _{\ell = 0}^{n_{j}1} Q_{j,\ell }\right) T_{\mathbb {C},2,\alpha }. \end{aligned}$$
 (e)For \(j \in \mathbb {N}\), denote the prime ideal \(P_j \cap S_{\mathbb {C},2,\alpha }\) of \(S_{\mathbb {C},2,\alpha }\) by \(P_j'\). Thus \(P_j' = (\bigcap _{\ell = 0}^{n_{j}1} Q_{j,\ell })S_{\mathbb {C},2,\alpha }\). For each \(j \in \mathbb {N}\), (writing \(\theta \) also for the image of \(\theta \) in \(S_{\mathbb {C},2,\alpha }/P_j'\) and in \(T_{\mathbb {C},2,\alpha }/P_j\)),In particular, \(T_{\mathbb {C},2,\alpha }/P_j\) is an Azumaya algebra over its centre \(\mathbb {C}[\theta ^{\pm n_j}]\), so the maximal ideals of \(T_{\mathbb {C},2,\alpha }/P_j\) are parametrised by \(\mathbb {C}\), and are the ideals$$\begin{aligned} (S_{\mathbb {C},2,\alpha }/P_j')[\theta ^{1}] = T_{\mathbb {C},2,\alpha }/P_j \quad \cong M_{n_j}(\mathbb {C}[\theta ^{\pm n_j}]). \end{aligned}$$(14)Hence, for all \(j \in \mathbb {N}\) and \(\lambda \in \mathbb {C}\),$$\begin{aligned} \{ M_{j, \lambda }/P_j : \lambda \in \mathbb {C}\}. \end{aligned}$$$$\begin{aligned} T_{\mathbb {C},2,\alpha }/M_{j, \lambda } \quad \cong \quad M_{n_j}(\mathbb {C}). \end{aligned}$$
 (f)
The integers \(n_j\), for \(j \in \mathbb {N}\), are unbounded; more precisely, every sufficiently large prime number occurs amongst the \(n_j\).
Proof
The partition of \(\mathrm {Spec}(S_{\mathbb {C},2,\alpha })\) into \(\mathcal {V}(\theta )\) and \(\mathcal {C}(\theta )\) is clear, and (a) and (b) follow at once since \(S_{\mathbb {C},2,\alpha }/\theta S_{\mathbb {C},2,\alpha } \cong R\) and \(T_{\mathbb {C},2,\alpha }\) is the localization of \(S_{\mathbb {C},2,\alpha }\) with respect to powers of \(\theta \).
By [11, Theorem 3.1], \(\mathbb {C}[x,y]\) has countably infinitely many maximal ideals with a finite \(\langle \alpha \rangle \)orbit. Enumerating these orbits by the parameter \(j \in \mathbb {N}\), and letting \(n_j\) be the size of the jth orbit, the integers \(n_j\) satisfy (f), by [11, Corollary 8.6 and note added in proof, page 97]. Labelling the jth finite \(\langle \alpha \rangle \)orbit as in (d), and defining the corresponding induced ideal \(P_j\) also as in (d), yields a countably infinite set of comaximal prime ideals of \(T_{2,\alpha , \mathbb {C}}\) by Lemma 2.3(b), giving the subset (c)(ii) of \(\mathrm {Spec}(T_{\mathbb {C},2,\alpha })\), see also [45, Theorem III.31]. The subset (c)(iii) is clear.
It remains to prove (e), in the course of which the maximal ideals of (c)(i) will be described. It follows easily from Lemma 2.3(b) that \(T_{\mathbb {C}, 2, \alpha }/P_j\quad \cong \quad {{\overline{R}}}[\theta ^{\pm 1}; \alpha ]\) where \({{\overline{R}}}\) is the direct sum of \(n_j\) copies of \(\mathbb {C}\) and \(\alpha \) acts on \({\overline{R}}\) by \(\alpha (e_i)=e_{(i+1)\mod n_j}\), where \(\{e_i: 1\le i\le n_j\}\) is the set of primitive idempotents of \({{\overline{R}}}\). Set \(n=n_j\), \(f=(1e_1)\theta , a=\theta ^{n+1}\) and \(b=\theta ^{1}\). Then: \(f^{n1}=(1e_1)(1e_2)\ldots (1e_{n1}) \theta ^{n1}=e_n\theta ^{n1}\), \(af^{n1}=\theta ^{n+1}e_n\theta ^{n1}=e_1\) and \(fb=1e_1\). Thus \( af^{n1}+fb=1\) and \(f^n=0 \). Therefore, by [1, Theorem 1.3], \(R[\theta ^{\pm 1};\alpha ]=M_n(B)\) with \(e_{11}=af^{n1}=e_1\). As \(\theta ^n\) is central and \(e_1\theta ^ke_1=0\), for \(0<k<n\), we have \(B=e_{11}R[\theta ^{\pm 1};\alpha ]e_{11}=(e_1R)[\theta ^{\pm n}]\simeq \mathbb {C}[\theta ^{\pm n}]\) and (14) holds. Now using standard arguments it is easy to complete the proof. \(\square \)
We can now summarise our (at present incomplete) knowledge about the occurrence of property \((\diamond )\) for the algebras \(S_{\mathbb {C},2,\alpha }\). Recall that, by definition, every element of \(\mathrm {Aut}_{\mathbb {C}\mathrm {alg}}(\mathbb {C}[x,y])\) is either triangular or square.
Theorem 7.4
 (a)
Suppose that \(\alpha \) is triangular. Then \(S_{\mathbb {C},2,\alpha }\) satisfies \((\diamond )\) if and only if \(\alpha \) has finite order if and only if \(\alpha \) is conjugate to an element of type (i), with \(\lambda , \mu \) roots of unity in \(\mathbb {C}\).
 (b)
Suppose that \(\alpha \) is square. Then \(S_{\mathbb {C},2,\alpha }\) satisfies \((\diamond )\) if and only if \(S_{\mathbb {C},2,\alpha }\) is not primitive.
Proof
 (a)
This is part of Lemma 7.2.
 (b)
Suppose that \(S_{\mathbb {C},2,\alpha }\) is primitive. Then \(S_{\mathbb {C},2,\alpha }\) does not satisfy \((\diamond )\) by Theorem 5.4(b). Suppose on the other hand that \(S_{\mathbb {C},2,\alpha }\) is not primitive. Then, from the description of the prime spectrum of \(S_{\mathbb {C},2,\alpha }\) in Lemma 7.3, it is clear, bearing in mind Kaplansky’s theorem [4, Theorem I.13.3], that the only primitive ideals of \(S_{\mathbb {C},2,\alpha }\) are the coArtinian maximal ideals \(M_{j,\lambda }\). Hence, every simple \(S_{\mathbb {C},2,\alpha }\)module is finite dimensional over \(\mathbb {C}\). Therefore, \(S_{\mathbb {C},2,\alpha }\) satisfies \((\diamond )\) by Theorem 6.1.
Of course, Theorem 7.4(b) begs the following obvious question. Keep the notation of Theorem 7.4, and suppose that \(\alpha \) is a square automorphism. Is \(S_{\mathbb {C},2,\alpha }\) primitive? The simplest square automorphism is perhaps the map \(x \mapsto y,\, y\mapsto x + y^2\). We do not even know the answer in this case.
Using the Leroy–Matczuk criterion for primitivity, Theorem 4.2, coupled with Lemma 7.3, it is not hard to reduce the above question to one about the geometry of the finite orbits of square automorphisms, a question which may be of independent interest. It seems more natural to frame it in terms of points in the plane \(\mathbb {C}^2\), rather than in terms of maximal ideals of R:
Proposition 7.5
Keep the notation of Theorem 7.4, with \(\alpha \) square. Denote the countably infinitely many finite \(\langle \alpha \rangle \)orbits of points in \(\mathbb {C}^2\) by \(\mathcal {P}_j\), for \(j \in \mathbb {N}\). Then \(S_{\mathbb {C},2,\alpha }\) is primitive if and only if a point \((a_j,b_j)\in \mathcal {P}_j\) can be chosen, for every \(j \in \mathbb {N}\), such that \(\{(a_j,b_j) : j \in \mathbb {N}\}\) lies on a (not necessarily irreducible) affine curve in \(\mathbb {C}^2\).
We leave the straightforward proof to the reader and refer to [25, Lemma 2.6(v)] for an equivalent description of \(\alpha \)special rings.
7.3 Subalgebras of group algebras over absolute fields
This family of examples shows that neither of the conditions (c) Ssatisfies a polynomial identity nor (d) \(\alpha  < \infty \) is implied by conditions (a) or (b) of Theorem 6.1. Note that (c) and (d) are equivalent when R is semiprime, by [9] or [35].
Let S be the subalgebra \(kA[\theta ;\alpha ]\) of kG. We claim that S inherits from kG the property that all its simple modules are finite dimensional. To see this, let W be a simple Smodule. Since the powers of \(\theta \) form an Ore set in S, W is either \(\theta \)torsion or \(\theta \)torsion free. In the first case, \(W\theta = 0\), so that W is a simple module over \(S/\theta S \cong kA\). Therefore \(\mathrm {dim}_k(W) < \infty \) by Hilbert’s Nullstellensatz. In the second case, \(W\theta = W\), so W carries a structure as \(S\langle \theta ^{1}\rangle \)module, that is as kGmodule. As such, it is necessarily simple, and hence has finite kdimension by Roseblade’s theorem [38].
Finally, choosing M to be a matrix of infinite order, we obtain an algebra S which is a skew polynomial kalgebra over a Laurent polynomial coefficient algebra, which has property \((\diamond )\) but for which \(\alpha  = \infty \).
7.4 A subalgebra of a group algebra in characteristic 0
The following example is considered in [25, 7.10–7.14].
Let \(S=R[\theta ; \alpha ]\), where \(R=\mathbb {C}[x^{\pm 1},y^{\pm 1}]\) and \(\alpha \in \mathrm {Aut}_{\mathbb {C}\mathrm {alg}}(R)\) is defined by \(\alpha (y)=x\) and \(\alpha (x)=yx^{1}\). Note that \(\alpha  = \infty \). In [25], Jordan studies both the algebras \(S := R[\theta ;\alpha ]\) and \(T := R[\theta ^{\pm 1}; \alpha ]\). He proves in [25, Proposition 7.13] that T is primitive.
In the same proposition it is claimed that R is not \(\alpha \)special, so that, by the Leroy–Matczuk theorem, Theorem 4.2, S is therefore not primitive. As is shown in [25, Propositions 7.11, 7.12], every nonzero primitive ideal of S is coArtinian. Hence, if S itself is not primitive, then every simple Smodule is finite dimensional, and S satisfies \((\diamond )\) by Theorem 6.1. However, as we observed in the first version of the current paper, there is a gap in the proof that R is not \(\alpha \)special in [25, Proposition 7.13]. We are indebted to Ken Goodearl and to the referee, who independently provided arguments to repair the gap in Jordan’s work. We give Goodearl’s argument in Theorem 7.6. It is shorter than the referee’s proof, although requiring the application of a result from [3]. The referee’s argument is sketched in Remark 7.8. We thank both Goodearl and the referee for their input, and for permission to include the arguments below.
Theorem 7.6
(Goodearl [15]) Let \(\alpha \) be the automorphism of the variety \({(\mathbb {C}^{\times })}^2\) given by \(\alpha (a,b)=(b,ab)\). There does not exist any curve \(C\subseteq {(\mathbb {C}^{\times })}^2\) which meets every finite \(\alpha \)orbit.
Proof
Assume there does exist a curve \(C\subseteq {(\mathbb {C}^{\times })}^2\) which meets every finite \(\alpha \)orbit.
As Jordan showed in [25, Proposition 7.11(ii)], for each prime \(p\equiv 1\, \hbox {(mod 5)}\), there exists a finite \(\alpha \)orbit \(\mathcal{O}_p\) consisting of points \((\omega ^{k^m}_p, \omega _p^{k^{m+1}})\) for \(m\in \mathbb {N}\), where k is an integer satisfying \(k^2\equiv k+1\, \hbox {(mod p)}\) and \(\omega _p\) is a primitive pth root of unity in \(\mathbb {C}\). In particular, both coordinates of every point in \(\mathcal{O}_p\) are primitive pth roots of unity. By Dirichlet’s Theorem, there are infinitely many primes \(p\equiv 1\, \hbox {(mod 5)}\), so Open image in new window is infinite. Hence, there is at least one irreducible component \(C_i\) of C such that Open image in new window is infinite.
For any prime \(p\equiv 1\, \hbox {(mod 5)}\), the orbit \(\mathcal{O}_p\) is contained in the finite subgroup \(H_p:=\{(\omega ^i_p,w^j_p) i,j\in \mathbb {Z}\}\) of \({(\mathbb {C}^{\times })}^2\). Consequently, there are infinitely many primes \(p\equiv 1\, \hbox {(mod 5)}\) such that \(C_i\cap (H_p\backslash \{(1,1)\})\) is nonempty. Thus, the intersection of \(C_i\) with the union of the finite subgroups of \({(\mathbb {C}^{\times })}^2\) is infinite. By [3, Corollary 1], \(C_i\) is contained in a proper algebraic subgroup \(\Gamma \) of \({(\mathbb {C}^{\times })}^2\).
Identify \(\mathbb {C}[x^{\pm 1}, y^{\pm 1}]\) with \(\mathcal{O}({(\mathbb {C}^{\times })}^2)\). Since \(\Gamma \) is a proper closed subgroup of \({(\mathbb {C}^{\times })}^2\), there exist integers s and t, not both 0, such that \(\Gamma \subseteq V(1x^sy^t)\). Hence, there are infinitely many primes \(p\equiv 1\, \hbox {(mod 5)}\) such that \(V(1x^sy^t)\cap \mathcal{O}_p\) is nonempty.
As claimed in the proof of [25, Proposition 7.13] (or as in Proposition 7.5), the non\(\alpha \)speciality in this example can be stated in geometric terms as in Theorem 7.6. Taking into account the previous discussion it follows that:
Corollary 7.7
Let \(R=\mathbb {C}[x^{{\pm }1}, y^{{\pm }1}]\) and \(\alpha \in \mathrm {Aut}_{\mathbb {C}\mathrm {alg}}(R)\) defined by \(\alpha (y)=x\) and \(\alpha (x)=yx^{1}\) and \(S=R[\theta ; \alpha ]\). Then S is not primitive. Moreover, every simple Smodule is finite dimensional, whereas \(\alpha  = \infty \). In particular, S satisfies \((\diamond )\), but does not satisfy a polynomial identity.
Remark 7.8
Here is the promised sketch of the referee’s patch for [25, Proposition 7.13]. With the help of the Nullstellensatz one can reduce to the case when the coefficients of the \(\alpha \)special element f belong to a finite Galois extension of \(\mathbb {Q}\), then, replacing f by \(\prod _{\sigma \in G} f^{\sigma }\) where G is the Galois group of the field extension, we may assume that f has coefficients in \(\mathbb {Q}\). Think of \(\alpha \) as acting by the matrix \(M=\left[ \begin{array}{cc}1 &{} 1\\ 1 &{} 0\end{array}\right] \). By Dirichlet’s Theorem and quadratic reciprocity there are infinitely many primes p such that the polynomial \(\mathrm {det}(M\lambda I_3)\) splits mod p. For such a prime p we can choose for M an eigenvalue \(c+p\mathbb {Z}\in \mathbb {Z}/p \mathbb {Z}\) and an eigenvector \((b_1+p\mathbb {Z}, b_2+p\mathbb {Z})\) in \(\mathbb {Z}/p\mathbb {Z}\times \mathbb {Z}/p\mathbb {Z}\). Now, for \(\omega _p\) a primitive pth root of unity, it follows that \(\alpha ^{1}(\omega ^{b_1}_p,\omega ^{b_2}_p)=(\omega ^{b_1c}_p,\omega ^{b_2c}_p)\). Thus the \(\alpha \)orbit of \((\omega ^{b_1}_p,\omega ^{b_2}_p)\) is contained in its Galois orbit. So f(x, y) vanishes on these orbits. Therefore, for infinitely many primes p there is at least one finite \(\alpha \)orbit \(\mathcal{O}_p\) of a point \((\omega _p^e, \omega _p^g)\) with e, g not both zero (mod p) such that f(x, y) vanishes on every point of \(\mathcal{O}_p\). Let X be the Zariski closure of the union of those infinitely many \(\mathcal{O}_p\) and \(X_1, \ldots , X_n\) its one dimensional irreducible components. Since \(\alpha \) permutes the \(X_i\), for each i there is an integer \(m\ge 1\) such that each \(X_i\) is stable under \(\alpha ^m\). Also as \(X_i\) has a dense set of points that are periodic under \(\alpha ^m\), \(\alpha ^m\) has finite order on \(X_i\). Let \(N\ge 1\) be such that \(\alpha ^N\) is the identity on every \(X_i\). Now \(\alpha ^N\) is the identity on each of the \(\mathcal{O}_p\) for infinitely many p. Hence 1 is an eigenvalue of \(M^N\), a contradiction.
8 Questions
For the convenience of the reader, we gather here a number of open questions arising from this work, some of them previously raised earlier in the text, some appearing here for the first time. As usual, k is a field, and S will denote the skew polynomial ring \(R[\theta ; \alpha ]\), where R is commutative Noetherian and \(\alpha \) is an automorphism. Further hypotheses will be added as needed.
First, a question which seeks to remove the gap between the necessary and sufficient conditions in Theorem 1.1 (= 5.4).
Question 8.1
Suppose that S is primitive and satisfies \((\diamond )\). Must R be a finite direct sum of fields?
We should note that the remaining case of Theorem 5.4 is the one where \(\alpha \) is of infinite order, and R is an \(\alpha \)special Noetherian domain of Krull dimension 1 with countable spectrum. Moreover \(R{\mathcal A}^{1}\) is a field, where \({\mathcal A}\), as before, is the smallest \(\alpha \)stable Ore set of S containing an \(\alpha \)special element. When \({\mathcal A}\) is finitely generated as a multiplicatively closed set (it is then possible to replace \({\mathcal A}\) by the set of powers of an \(\alpha \)special element) R is a Gdomain (see [26, Theorem 19]). Thus, by [26, Theorem 156], R is semilocal. The case when \(R=K[[X]]\), the ring of formal power series and \(\alpha \) is the automorphism given by \(\alpha (X)=qX\), for \(q\in K\) not a root of unity, is a key example to consider.
A second point where necessary and sufficient conditions fail to match is in the affine case, where the proof in one direction of Theorem 1.2 (= 6.1) needs the base field to be uncountable or of characteristic zero (cf. Proposition 5.3’). So we ask:
Question 8.2
Suppose that R is an affine algebra over a countable field k of nonzero characteristic and S satisfies \((\diamond )\). Are all simple Smodules finite dimensional over k?
Haunting the results of Sect. 6 and the examples of Sect. 7 is the fundamental, if vague, question:
Question 8.3
Suppose that R is kaffine and \(\alpha \in \mathrm {Aut}_{k\mathrm {alg}}(R)\). Are there “reasonable” necessary and sufficient hypotheses on R and \(\alpha \) which determine when all the simple Smodules are finitedimensional?
The examples stemming from group algebras considered in Sect. 7.3 show that the situation regarding Question 8.3 in positive characteristic is undoubtedly rather delicate. In characteristic 0, Sect. 7.4 provides an example where S is kaffine, has all its simple modules finite dimensional but does not satisfy a polynomial identity. Further such examples may be provided by the square automorphisms of \(\mathbb {C}[x,y]\) studied in Sect. 7.2, where the question of primitivity remains open. We thus repeat the question asked after Theorem 7.4.
Question 8.4
Let \(S = S_{\mathbb {C},2,\alpha } = \mathbb {C}[x,y][\theta ; \alpha ]\), where \(\alpha \) is a square automorphism. For example, \(\alpha \) could be the map sending x to y and y to \(x + y^2\). Is S primitive? Equivalently, does S have an infinite dimensional simple module?
Footnotes
 1.
 2.
[11, Theorem 2.6] is slightly misstated, since the cyclically reduced automorphisms which it treats include all the square automorphisms, but also the affine automorphisms which are not also elementary, such as, for example the map \(\tau : x\mapsto y, \, y\mapsto x\). See the definitions on pages 68–69 of [11]. In fact, [11, Theorem 2.6] is valid, in the terminology of that paper, for the elements of A having degree at least 2.
Notes
Acknowledgements
Some of this work was done while the second author visited the University of Glasgow and the University of Warsaw, supported by Grant SFRH/BSAB/1286/2012. She would like to thank the two universities for their hospitality. The second author was also partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds (FEDER), under the partnership agreement PT2020 and by the Warsaw Center of Mathematics and Computer Science. The first and third authors would like to thank the University of Porto and its staff for hospitality and good working conditions. The third author acknowledges the support of CMUP and of Polish National Center of Science Grant no. DEC2011/03/B/ST1/04893. The first author was supported by Leverhulme Fellowship EM2017081/9. We are thankful to David Jordan, Christian Lomp, António Machiavelo and Carlos Rito for very helpful discussions regarding the behaviour of orbits in \(\mathbb {C}^2\) under the action of automorphisms. We are particularly grateful to Jason Bell, Ken Goodearl and the referee for their careful reading, for many helpful comments, and for their proofs of Theorem 7.6, and for the remarks following Theorem 6.1 on the extension of that result to countable fields.
References
 1.Agnarsson, G., Amitsur, S.A., Robson, J.C.: Recognition of matrix rings. II. Isr. J. Math. 96, 1–13 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
 2.Bell, J.: Private communicationGoogle Scholar
 3.Bombieri, E., Masser, D., Zannier, U.: On unlikely intersections of compex varieties with tori. Acta Arith. 133, 309–323 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
 4.Brown, K.A., Goodearl, K.R.: Lectures on Algebraic Quantum Groups. Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel (2002)Google Scholar
 5.Brown, K.A., Warfield Jr., R.B.: The influence of ideal structure on representation theory. J. Algebra 116(2), 294–315 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
 6.Carvalho, P.A.A.B., Lomp, C., PusatYilmaz, D.: Injective modules over downup algebras. Glasg. Math. J. 52(A), 53–59 (2010)Google Scholar
 7.Carvalho, P.A.A.B., Hatipoğlu, C., Lomp, C.: Injective hulls of simple modules over differential operator rings. Commun. Algebra 43(10), 4221–4230 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
 8.Carvalho, P.A.A.B., Musson, I.M.: Monolithic modules over Noetherian rings. Glasg. Math. J. 53(3), 683–692 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
 9.Damiano, R.F., Shapiro, J.: Twisted polynomial rings satisfying a polynomial identity. J. Algebra 92(1), 116–127 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
 10.Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150. Springer, Berlin (2004)Google Scholar
 11.Friedland, S., Milnor, J.: Dynamical properties of plane polynomial automorphisms. Ergodic Theory Dyn. Syst. 9, 67–99 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
 12.Formanek, E., Jategaonkar, A.V.: Subrings of Noetherian rings. Proc. Am. Math. Soc. 46(2), 181–186 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
 13.Gilmer, R.: Multiplicative Ideal Theory, Queen’s Papers in Pure and Applied Mathematics, vol. 90 (1992)Google Scholar
 14.Goodearl, K.R.: Classical localizability in solvable enveloping algebras and PoincaréBirkoffWitt extensions. J. Algebra 132, 324–377 (1990)MathSciNetCrossRefGoogle Scholar
 15.Goodearl, K. R.: Private communicationGoogle Scholar
 16.Goodearl, K.R., Warfield Jr., R.B.: An Introduction to Noncommutative Noetherian Rings, LMS Student Texts 61. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
 17.Hatipoğlu, C., Lomp, C.: Injective hulls of simple modules over finite dimensional nilpotent complex Lie superalgebras. J. Algebra 361, 79–91 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 18.Hall, P.: On the finiteness of certain soluble groups. Proc. Lond. Math. Soc. (3) 9, 595–622 (1959)Google Scholar
 19.Hirano, Y.: On injective hulls of simple modules. J. Algebra 225(1), 299–308 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
 20.Hrushowski, E.: The Elementary Theory of the Frobenius Automorphism. http://www.ma.huji.ac.il/~ehud/FROB.pdf
 21.Irving, R.S.: Prime ideals of ore extensions over commutative rings. J. Algebra 56, 315–342 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
 22.Jategaonkar, A.V.: Jacobson’s conjecture and modules over fully bounded Noetherian rings. J. Algebra 30, 103–121 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
 23.Jategaonkar, A.V.: Integral group rings of polycyclicbyfinite groups. J. Pure Appl. Algebra 4, 337–343 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
 24.Jategaonkar, A.V.: Localization in Noetherian rings. London Math. Soc. Lecture Note Series, vol. 98. Cambridge University Press, Cambridge (1986)Google Scholar
 25.Jordan, D.A.: Primitivity in skew Laurent polynomial rings and related rings. Math. Z. 213(3), 353–371 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
 26.Kaplansky, I.: Commutative Rings, Revised edition. The University of Chicago Press Ltd., Chicago (1974)zbMATHGoogle Scholar
 27.Karamzadeh, O.A.S., Moslemi, B.: On Gtype domains. J. Algebra Appl. 11(1), 1250005 (2012)Google Scholar
 28.Lane, D.R.: PhD thesis, University of London (1976)Google Scholar
 29.Leroy, A., Matczuk, J.: Primitivity of skew polynomial and skew Laurent rings. Commun. Algebra 24(7), 2271–2284 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
 30.Matlis, E.: Injective modules over Noetherian rings. Pacif. J. Math. 8, 511–528 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
 31.McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings. Graduate Studies in Mathematics, vol. 30. American Mathematical Society, Providence, RI (2001)Google Scholar
 32.Musson, I.M.: Injective modules for group rings of polycyclic groups, II. Q. J. Math Oxford Ser. 2(31), 449–466 (1980)CrossRefzbMATHGoogle Scholar
 33.Musson, I.M.: Some examples of modules over Noetherian rings. Glasg. Math. J. 23, 9–13 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
 34.Musson, I.M.: Finitely generated, nonArtinian monolithic modules. In: New trends in Noncommutative Algebra (Seatle, WA, 2010) Contemp. Math., Amer. Math. Soc., , vol. 562, pp. 211–220. ProvidenceGoogle Scholar
 35.Pascaud, J.L., Valette, J.: Polynomes tordue a identite polynomiale. Commun. Algebra 16, 2415–2425 (1988)CrossRefzbMATHGoogle Scholar
 36.Poole, D.G.: Localization in ore extensions of commutative noetherian rings. J. Algebra 128, 434–445 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
 37.Tanase, R.: Complex Hénon maps and discrete groups. Adv. Math. 295, 53–89 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
 38.Roseblade, J.E.: Group rings of polycyclic groups. J. Pure Appl. Algebra 3, 307–328 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
 39.Roseblade, J.E.: Applications to the ArtinRees lemma to group rings, Sympos. Math. vol. 17, pp. 471–478 (convegno sui Gruppi Infiniti, INDAM, Rome 1973. Academic Press, London (1976)Google Scholar
 40.Sant’ana, A., Vinciguerra, R.: On Cyclic Essential Extensions of Simple Modules Over Differential Operator Rings. arXiv:1704.04970v1
 41.Sharp, R.Y., Vámos, P.: Baire’s category theorem and prime avoidance in complete local rings. Arch. Math. 44, 243–248 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
 42.Smith, L.: Polynomial Invariants of Finite Groups, Research Notes in Mathematics, 6. A K Peters, Wellesley (1995)CrossRefGoogle Scholar
 43.Smith, M.: Eigenvectors of automorphisms of polynomial rings in two variables. Houst. J. Math. 10, 559–573 (1984)MathSciNetzbMATHGoogle Scholar
 44.Stenström, B.: Rings of Quotients. Springer, Berlin (1975)CrossRefzbMATHGoogle Scholar
 45.Zariski, O., Samuel, P.: Commutative Algebra. D. Van Nostrand Company Inc, London (1958)zbMATHGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.