Pointwise minimal extensions

We characterize pointwise minimal extensions of rings introduced by P.-J. Cahen, D. E. Dobbs and T. G. Lucas.


Introduction and Notation
We consider the category of commutative and unital rings and its epimorphisms. A local ring is here what is called elsewhere a quasilocal ring. As usual, Spec(R) and Max(R) are the sets of prime and maximal ideals of a ring R. The characteristic of an integral domain k is denoted by c(k). Finally, ⊂ denotes proper inclusion, |X| the cardinality of a set X and P the set of all prime numbers.
The conductor of a (ring) extension R ⊆ S is denoted by (R : S), the set of all R-subalgebras of S by [R, S] and the integral closure of R in S by R. Any writing [R, S] is relative to some extension R ⊆ S. Clearly, ([R, S], ⊆) is a lattice since it is stable under the formation of arbitrary intersections (meets) and compositums (joins). If [R, S] has some property P of lattices, we say that R ⊆ S has the property P.
An extension R ⊆ S is called an afffine pair (or strongly affine) if each R-subalgebra of S is of finite type. We say that an extension R ⊆ S is finite if the R-module S is finitely generated.
The extension R ⊆ S is said to have FIP (or is called an FIP extension) (for the "finitely many intermediate algebras property") if [R, S] is finite. A chain of R-subalgebras of S is a set of elements of [R, S] that are pairwise comparable with respect to inclusion. We say that an extension R ⊆ S has FCP (or is called an FCP extension) (for the "finite chain property") if each chain in [R, S] is finite. Dobbs and the authors characterized FCP and FIP extensions [7]. A mighty tool is the concept of minimal (ring) extensions, introduced by Ferrand-Olivier [11]. Recall that an extension R ⊂ S is called minimal if [R, S] = {R, S}.
The key connection between the above ideas is that if R ⊆ S has FCP, then any maximal (necessarily finite) chain of R-subalgebras of S, R = R 0 ⊂ R 1 ⊂ · · · ⊂ R n−1 ⊂ R n = S, with length n < ∞, results from juxtaposing n minimal extensions R i ⊂ R i+1 , 0 ≤ i ≤ n − 1. We come now to the subject of the paper. In [4], Cahen, Dobbs and Lucas call an extension R ⊂ S pointwise minimal if R ⊂ R[t] is a minimal extension for each t ∈ S \ R. We study such extensions in Section 3 and a special type of these extensions: a ring extension R ⊂ S is called a pointwise minimal pair if T ⊂ S is pointwise minimal for each T ∈ [R, S] \ {S}.
Clearly, the following implications hold: minimal extension ⇒ pointwise minimal pair ⇒ pointwise minimal extension. We also define a dual notion in Section 3; that is, co-pointwise minimal extensions. In [20], we called an extension which is a minimal flat epimorphism, a Prüfer minimal extension. From now on, we use this terminology. A pointwise minimal extension is either integral or integrally closed, in which case it is Prüfer minimal. It follows that our study can be reduced to the case of integral extensions. A pointwise minimal extension R ⊂ S has a crucial ideal M i.e. the support of the R-module S/R is {M} and M is necessarily a maximal ideal. In case R ⊂ S is integral and pointwise minimal, its crucial ideal is (R : S). Those statements (appearing in [4] in a special context) are proved in Section 2 and Section 3 and are essential in this paper. We will also need the canonical decomposition of an integral extension R ⊆ + S R ⊆ t S R ⊆ S, where + S R and t S R are the seminormalization and the t-closure of R in S (see Section 2 for the details). Our strategy is as follows . We first suppose that R is a field in Section 4 and then we consider in Section 5 an integral extension, whose conductor is a maximal ideal, much more easy to handle than a crucial ideal. Surprisingly, we are able to classify pointwise minimal integral extensions R ⊂ S: either the seminormalization and the t-closure coincide or R = + S R and t S R = S. Then in Section 5 we get a complete characterization of pointwise minimal integral extensions and pairs that are not minimal, while co-pointwise minimal extensions are characterized in Section 3 as pointwise minimal pairs of length 2 and more precisely at the end of Section 5. Naturally, we consider the special case of FCP and FIP extensions. Section 6 is concerned with examples and applications. In particular, we consider Nagata extensions. To end, Section 7 deals with properties of lattices and their atoms (in our context, they are minimal extensions), linked to the above notions. In particular, finitely geometric lattices are involved.

Some useful results and recalls
We need to give some notation and definitions. If I an ideal of a ring R, we denote by V R (I) or (V(I)) the closed subset {P ∈ Spec(R) | I ⊆ P }, by D R (I) its complement and by R √ I the radical of I in R. The support of an R-module E is Supp R (E) := {P ∈ Spec(R) | E P = 0}, and MSupp R (E) := Supp R (E) ∩ Max(R). If R ⊆ S is a ring extension and P ∈ Spec(R), then S P is both the localization S R\P as a ring and the localization at P of the R-module S. For a ring morphism f : R → S and Q ∈ Spec(S), we denote by κ(P ) → κ(Q) the residual extension, where P = f −1 (Q).
Definition 2. 1. We say that an extension R ⊂ S has a a crucial ideal C(R, S) := M ∈ Spec(R) if Supp R (S/R) = {M} and in this case call the extension M-crucial. A crucial ideal needs to be maximal because a support is stable under specialization.
For example, a minimal extension has a crucial ideal [11, Théorème 2.2]. We will show later that a pointwise minimal extension has also a crucial ideal. We begin by proving some results on crucial ideals.

2.1.
Crucial ideals and radicial extensions. In the sequel, {R α | α ∈ I} is the family of all finite extensions R ⊂ R α with R α ∈ [R, S] and conductor C α .
Proposition 2.2. Let R ⊂ S be an extension, with conductor C. The following statements hold: (1) If R ⊂ S is M-crucial, then C ⊆ M.
(2) If R ⊂ S is integral, then R ⊂ S has a crucial ideal if and only if √ C ∈ Max(R), and then C(R, S) = √ C.

Proof.
(1) If the extension is M-crucial, suppose that there is some x ∈ C \ M, then it is easily seen that R M = S M , a contradiction.
(2) For M ∈ Spec(R), observe that M is a crucial ideal of R ⊂ S if and only if M is a crucial ideal of each R ⊂ R α . Then it is enough to use the following facts: An M-crucial integral extension has the following properties. If Q ∈ Spec(S) is lying over P ∈ Spec(R), then R P → S P and κ(P ) → κ(Q) The Nagata ring of a ring R is R(X) := R[X] Σ , where Σ is the multiplicatively closed subset of polynomials whose contents are R. We compute the crucial ideal of a Nagata extension R(X) Proof. The extension g : R → R(X) is faithfully flat and Supp(S/R) = {M}. Let Q ∈ Supp(S(X)/R(X)). Applying [8,Proposition 2.4 (b)], we get that Q ∈ ( a g) −1 (Supp(S/R)), so that a g(Q) = Q ∩ R ∈ Supp(S/R) = {M}, giving M = Q ∩ R. It follows that M ⊆ Q, which implies MR(X) ⊆ Q and then Q = MR(X) since MR(X) ∈ Max(R(X)). Therefore, Supp(S(X)/R(X)) = {MR(X)}.
We will call in this paper radicial any purely inseparable field extension, in order to have a terminology consistent with radicial (radiciel in French) extensions of rings. Recall that a ring morphism R → S is called radicial if Spec(R ′ ⊗ R S) → Spec(R ′ ) is injective for any base change R → R ′ . A ring extension R → S is radicial if and only if Spec(S) → Spec(R) is injective and its residual extensions are radicial [13,Proposition 3.7.1]. Also a radicial extension K ⊂ L of fields is said to have height one if x p ∈ K for each x ∈ L, where p := c(K) ∈ P. We say that a ring extension K ⊂ S, where K is a field, is radicial of height one if c(K) = p ∈ P and x p ∈ K for each x ∈ S. Indeed, such an extension is radicial, as it is easily seen.
An M-crucial extension R ⊂ S, such that M = (R : S), is called a height one radicial extension if so is R/M ⊂ S/M. Such an extension is again radicial, by the above considerations.

2.2.
Results on minimal extensions. There are three types of minimal integral extensions, characterized by the following theorem, from the fundamental lemma of Ferrand-Olivier.
is minimal if and only if one of the following conditions is satisfied, in which case K ⊂ A is finite: (1) A is a field and K → A is a minimal field extension. ( Lemma 2. 5. The following statement hold: (1) Minimal field extensions coincide with minimal ring extensions between fields. (2) A minimal field extension is either separable or radicial. We give here a lemma used in earlier papers and introduce FMC extensions. An extension R ⊂ S is said to have FMC (for a "finite maximal chain" property) if there is a finite maximal chain of extensions going from R to S. Minimal and FCP extensions have FMC. Lemma 2. 7. Let R ⊂ S be an extension and T, U ∈ [R, S] such that R ⊂ T is a finite minimal extension and R ⊂ U is a Prüfer minimal extension. Then, C(R, T ) = C(R, U), so that R is not a local ring. Proof. Assume that C(R, T ) = C(R, U) and set M := C(R, T ) = (R : T ) = C(R, U) ∈ Max(R). Then, MT = M and MU = U because R ⊂ U is a Prüfer minimal extension. It follows that MUT = UT = MT U = MU = U, a contradiction.
be a finite maximal chain such that R 0 := R and

The canonical decomposition of an integral extension.
Definition 2. 9. An integral extension R ⊆ S is called infra-integral [16] (resp.; subintegral [23]) if all its residual extensions are isomorphisms (resp.; and the spectral map Spec(S) → Spec(R) is bijective). An extension R ⊆ S is called t-closed (cf. [16] If R ⊆ S is seminormal, then (R : S) is a radical ideal of S and of R (the proof goes back to Traverso). The seminormalization + S R of R in S is the smallest element B ∈ [R, S] such that B ⊆ S is seminormal and the greatest element The canonical decomposition of an arbitrary ring extension We note for further use that the classes of infra-integral and subintegral extensions are both stable under left or right divisions. The following proposition gives the link between the elements of the canonical decomposition and minimal extensions.
Moreover, if R ⊂ S is subintegral, (resp.; infra-integral, seminormal, t-closed), R ⊂ T has the same property. Proof. We can suppose that T = S. [19, Lemma 3.1] asserts that (2) and (5) holds. Now (1) is clear since we deal with a bijective spectral map. If R ⊂ S is seminormal, (R : S) is a finite intersection of maximal ideals of S (resp. R i+1 ) by an easy generalization of [7,Proposition 4.9], giving (3) and (4). 3. General properties on pointwise minimal extensions 3. 1. First results on pointwise minimal extensions. The next proposition generalizes to arbitrary extensions some results of [4] gotten in the integral domains context. The proofs need only slight changes. Conversely, assume that R ⊂ S is pointwise minimal. Let x, y ∈ S \ R, so that R ⊂ R[x] and R ⊂ R[y] are minimal, with respective crucial ideals M and N. Assume that N = M. For any P ∈ Spec(R) \ {M, N}, we have  It may be asked if the trichotomy: inert, ramified, decomposed of finite minimal extensions is still true for finite pointwise minimal extensions, in that sense: if some R[t] verifies some of these properties, all of them verify the same property. The answer is no (see Example 6.4(5)).  Proof. Assume that R ⊂ S is neither integrally closed nor integral. Pick x ∈ R \ R and y ∈ S \ R. The crucial ideal M of the extension is also the crucial ideal of R ⊂ R[x] and R ⊂ R[y]. The first one is minimal integral and the second one is Prüfer minimal since y ∈ R, which contradicts Lemma 2. 7 In their result, R is a non-integrally closed local integral domain, S is its quotient field and R ⊂ R is pointwise minimal. Then, each minimal overring of R is contained in R.
The integrally closed case gives a simple result.
We deduce from [7, Theorem 6.3] that R ⊂ S has FIP. To end, we get that ℓ[R, S] = 1, so that R ⊂ S is minimal by [7, Proposition 6.12].
Corollary 3. 7. A pointwise minimal extension is either Prüfer minimal or integral and these conditions are mutually exclusive. Proof. Let R ⊂ S be a pointwise minimal extension which is not integral. By Proposition 3.4, R ⊂ S is integrally closed. If R ⊂ S is integrally closed, then R ⊂ S is minimal Prüfer.
In order to characterize pointwise minimal integral extensions (resp.; pairs), next results will be useful.
Lemma 3. 8. Let R ⊂ S be an integral extension, with conductor C. The following statements hold: Assume that the supremum of the lengths ℓ[R, R α ] is a finite integer n. Then, there clearly exists some R β such that ℓ[R, R β ] = n. Assume now that R β = S, and let x ∈ S \ R β . Then, x is in some R γ and there exists some The last result is obvious.
In Section 4, we reduce our proofs to the case of fields. The following is enlightening. Consider a field extension K ⊂ L, which is pointwise minimal. This extension is necessarily algebraic, because a flat epimorphism whose domain is a field is surjective [20, Scholium A]. We will see later in Proposition 4.4 that K ⊂ L is either minimal separable or radicial of height one. Complexity of proofs relies heavily on this dichotomy.

3.2.
Co-pointwise minimal extensions. We are going to consider a property dual from the property of pointwise minimal extensions.
We can remark that the above definition without "x ∈ S \ R" is uninteresting because ipso facto this would mean that R ⊂ S is minimal. The next proposition shows that our definition of a co-pointwise minimal extension leads to a special case of pointwise minimal pairs. Proposition 3. 10. Let R ⊂ S be a ring extension. The following conditions are equivalent: (1) R ⊂ S is a co-pointwise minimal extension; (2) R ⊂ S is a pointwise minimal pair such that ℓ[R, S] = 2; (3) R ⊂ S is a pointwise minimal pair and the R-algebra S has a minimal system of generators whose cardinality is 2. In particular, a co-pointwise minimal extension has FCP.
In particular, a co-pointwise minimal extension has FCP.
(2) ⇒ (3) Assume that R ⊂ S is a pointwise minimal pair and ℓ[R, S] = 2. Then any maximal chain from R to S has length 2 and is of the In [10] and [6], Dobbs and Shapiro studied extensions of the form R ⊂ T ⊂ S, where R ⊂ T and T ⊂ S are minimal. Since a copointwise minimal extension has length 2, we may use their results. We will have more details about the connection with these papers after having characterized co-pointwise minimal extensions in Section 5.
Characterizations for arbitrary integral extensions will surprisingly lead to three special cases of the canonical decomposition.

The case of an integral extension over a field
For an ideal I of a ring and a positive integer n, we set I [n] := {x n | x ∈ I}. In this section, k ⊂ S is an integral extension and k is a field are the riding hypotheses. If k ⊂ L is an algebraic field extension and y ∈ L, the minimal polynomial of y over k is denoted by P k,y (X).
Proposition 4.1. If k ⊂ S is subintegral, then S is a local ring with maximal ideal N. Moreover, the following statements hold: Proof. Since k ⊂ S is subintegral, its spectal map is a homeomorphism so that S is a zero-dimensional local ring, with maximal ideal N. Then, (2) Assume that k ⊂ S is a pointwise minimal pair and let x, y ∈ N be two different elements. Then is minimal ramified with conductor kx. Then, xy ∈ kx gives that there is some a ∈ k such that xy = ax. The same reasoning gives some b ∈ k such that xy = by, so that ax = by which implies a = b = 0 = xy. Then, N 2 = 0.
Conversely, assume that  Proof.
(1) We have the following implications: k ⊂ S is minimal ⇒ k ⊂ S is a pointwise minimal pair ⇒ k ⊂ S is pointwise minimal. Now, assume that k ⊂ S is pointwise minimal and |k| = 2. By Lemma 3.8, S is the union of an upward directed family F of FCP extensions R α . For each α, there exists an integer n α = 0, 1 such that R α ∼ = k nα , since k ⊂ R α is a finite seminormal infra-integral extension by Lemma 4.2. We are going to show that n α = 2. Deny. Let e and f be two elements of the standard basis of the k-vector space k nα , so that e 2 = e, f 2 = f and ef = 0. It follows that {1, e, f } is free because n α > 2. Let λ = 0, 1 in k and set x = e + λf . Then, k ⊂ k[x] is minimal decomposed by Proposition 2.11 since x ∈ k. It follows that Then, there exist a, b ∈ k such that x 2 = a + bx, giving a + b(e + λf ) = e + λ 2 f , so that λ 2 = λ, a contradiction since λ = 0, 1. To conclude, we have n α = 2 and then k ⊂ R α is minimal and k ⊂ S is minimal by Lemma 3.8.
(2) Assume that |k| = 2 and let x ∈ S \ k. Since k ⊂ S is seminormal infra-integral and x is integral over k, we get that k ⊂ k[x] has FCP [7,Theorem 4.2] and k[x] ∼ = k n , for some integer n by Lemma 4.2, from which we infer that Assume that k ⊂ S is also a pointwise minimal pair. In view of Lemma 3.8, S is the union of an upward directed family F of FCP integral extensions R δ such that R δ ∼ = k n δ for some integer n δ , for each δ (see (1)). Assume that there exists some δ such that n δ ≥ 4, and let e α , e β and e γ be distinct elements of the standard basis of the k-vector space R δ . Set e := e α + e β , f := 1 + e and x := e α + e γ .
Conversely, assume that S ∼ = k n , with n ≤ 3. If n = 2, then S ∼ = k 2 and k ⊂ S is minimal and so a pointwise minimal pair. If n = 3, then S ∼ = k 3 and ℓ[k, S] = 2, so that k ⊂ S is a pointwise minimal pair. The last equivalence is then obvious.
Proposition 4.4. [4,Proposition 4.16] If k ⊂ S is t-closed, then S is a field and the following statements are equivalent: (1) k ⊂ S is a pointwise minimal extension; (2) k ⊂ S is a pointwise minimal pair; (3) k ⊂ S is either a minimal separable field extension, or a height one radicial extension. Proof We now intend to characterize pointwise minimal extensions k ⊂ S, where k is a field, that do not satisfy the conditions of Propositions 4.1, 4.3 and 4. 4. The following lemma gives a necessary condition on the seminormalization and the t-closure.
Lemma 4. 5. If k ⊂ S is pointwise minimal such that t S k = + S k, then, k ⊂ S is seminormal and infra-integral. Proof. In order to show that k ⊂ S is seminormal infra-integral, it is enough to show that k = + S k and S = t S k (Definition 2.9). So, deny that these two equations hold.
Assume first that k = + S k. Then, there exist x ∈ + S k \ k and y ∈ t S k \ + S k. We now observe that k ⊂ k[x] is minimal ramified and k ⊂ k[y] is minimal decomposed. Indeed, x ∈ + S k shows that k ⊂ k[x] is minimal ramified by Proposition 2.11(1). Moreover, y ∈ + S k shows that k ⊂ k[y] is not ramified and then is decomposed since y ∈ t S k, by Proposition 2.11(2). Set z := x + y.
is a field. Hence, y ∈ k, a contradiction. Then, S = t S k and k ⊂ S is infra-integral. The last case to consider is an extension of the form k ⊂ T ⊂ S, where k ⊂ T is subintegral and T ⊂ S is t-closed. (1) N [2] = 0.
(2) k ⊂ S is radicial of height one, whence also k ⊂ S/N. Such an extension is never a pointwise minimal pair. Proof. We get that (k : S) = 0 since k is a field. By Lemma 2.10, it follows that N := (T : S) is the only maximal ideal of T and S. Assume that k ⊂ S is pointwise minimal. Then so is k ⊂ T and (1) holds, since it satisfies the conditions of Proposition 4. 1.
We now show (2). Fix some x ∈ N \ k , so that x 2 = 0 by (1). Then, k ⊂ k[x] is minimal ramified by Proposition 2.11(1) because x ∈ + S k. Let y ∈ S \ T . It follows that k ⊂ k[y] is a minimal extension, which is neither ramified nor decomposed, since y ∈ t S k. So, k ⊂ k[y] is minimal inert and then t-closed, whence k[y] is a field by Proposition 4. 4. Now set z := x + y. We get that z ∈ t S k since x ∈ t S k and y ∈ t S k. Then, k ⊂ k[z] is minimal inert by the same proof used for k ⊂ k[y] and then a field extension by Proposition 4. 4. Let P (X) = P k,z (X) ∈ k[X], with derivative P ′ (X). From P (x + y) = 0, we deduce that P (y) + xP ′ (y) = 0. Since {1, x} is a basis of k[x, y] over the field k[y], we get P (y) = P ′ (y) = 0, from which we infer that P ′ (X) = 0, because P (X) is irreducible and P (y) = 0 shows that P (X) = P k,y (X). In particular, P (X) is not separable. Hence, k ⊂ k[y] is radicial by Lemma 2.5 and c(k) = p ∈ P. Now, k ∼ = T /N ⊂ S/N is a field extension. Ifȳ is the class of y in S/N, P (X) = P k,ȳ (X) and thenȳ is a radicial element. Now from T = k + N (k ⊂ T is subintegral), we deduce that each t ∈ T is of the form t = b + m, for some b ∈ k and m ∈ N. Then, t p = b p + m p = b p ∈ k by (1). If y belongs to S \ T , the above proof shows that the minimal extension k ⊂ k[y] is radicial and y p ∈ k by Lemma 2. 5. Then, k ⊂ S is radicial of height one, whence also k ⊂ S/N. The proof of (2) is now complete.
Conversely, assume that (1) and (2) hold. Proposition 4.1(1) entails that k ⊂ k[x] is minimal for each x ∈ T \ k. Let y ∈ S \ T andȳ its class in S/N. By (2), we get that y p = a ∈ k, givingȳ p = a ∈ k. From (2) we deduce that k ⊂ k[ȳ] is radicial and necessarily P k,ȳ = X p − a. It follows that X p − a = P k,y (X) and then k[y] ∼ = k[X]/(X p − a). Therefore, k[y] is a field such that [k[y] : k] = p and then k ⊂ k[y] is a minimal field extension. The proof is complete.
We claim that under these conditions, k ⊂ S is not a pointwise minimal pair. Take again x ∈ T \ k and y ∈ S \ T .
Remark 4. 7. We exhibit an extension k ⊂ S which satisfies the hypothesis of Proposition 4.6, such that N [2] = 0 and such that k ⊂ S/N is radicial of height one, but which is not pointwise minimal.
Let k be a field with c(k) = 2 and such that k = k [2] . Set R := k[T ]/(T 2 ) = k[t] = k + kt, where t, the class of T , satisfies t 2 = 0. Then, k ⊂ R is minimal ramified, so that R is a zero-dimensional local ring with maximal ideal M := kt. Pick some a ∈ k \ k [2] and set S : Then R ′ is a zero-dimensional local ring with maximal ideal N := kt + ktx, satisfying N [2] = 0 and R ⊂ R ′ is a minimal ramified extension such that M = (R : R ′ ) and R ′ /N ∼ = k. It is easy to check that N is an ideal of S, such that S/N ∼ = k + kx, where x is the class of x in S/N, and N = (R ′ : S). Moreover, x satisfies x 2 = a = a ∈ k, so that a). Therefore, N ∈ Max(S) and k ⊂ S/N is minimal radicial of height one. Then, R ′ ⊂ S is minimal inert and (S, N) is local. We infer that is not minimal.

Arbitrary integral extension
Gathering results of Propositions 4.1, 4.3, 4.4 and 4.6, we are now able to characterize pointwise minimal extensions and pointwise minimal pairs. We first give a statement for an integral extension k ⊂ S, where k is a field, and then for an arbitrary integral extension. As we saw in the previous sections, some pointwise minimal extensions are minimal. We first get rid of these cases in the next result.
Proposition 5. 1. Let R ⊂ S be an M-crucial extension, satisfying one of the following mutually exclusive conditions: Then, R ⊂ S is pointwise minimal if and only if R ⊂ S is minimal.

Proof. Use Propositions 3.6 for (1), 4.3 (1) for (2) and 4.4 for (3).
In the next result, we use Lemma 2.10 in conditions (1) and (2), because k ⊂ S is of the form k ⊆ T ⊆ S, where k ⊆ T is subintegral and T ⊆ S is t-closed. (1) t S k = + S k, N [2] = 0 and if t S k ⊂ S, then k ⊂ S is a height one radicial extension.
(3) |k| = 2 and k ⊂ S is a seminormal infra-integral extension. Conversely, assume that k ⊂ S is pointwise minimal. If t S k = + S k, then k ⊂ S is seminormal infra-integral by Lemma 4.5 and satisfies (3) by Proposition 4. 3. In particular, if k ⊂ S is a pointwise minimal pair, Proposition 4.3 shows also that k ⊂ S satisfies (4).
Assume now that t S k = + S k. If k = t S k = + S k = S, then k ⊂ S satisfies (1) by Proposition 4.6. Moreover, Proposition 4.6 says that k ⊂ S is not a pointwise minimal pair.
Two cases are remaining. The first one is when k = t S k = + S k = S, that is k ⊂ S is t-closed, and Proposition 4.4 gives (1) for both a pointwise minimal extension and a pointwise minimal pair. The second case is when k = t S k = + S k = S, that is k ⊂ S is subintegral, and Proposition 4.1 gives (1) for a pointwise minimal extension and (2) for a pointwise minimal pair.
Using the four conditions of Theorem 5.2, we are now able to give a complete characterization of pointwise minimal extensions and pairs. Theorem 5. 3. Let R ⊂ S be a non-minimal integral extension with M := (R : S) ∈ Max(R). Consider the following conditions: (3) |R/M| = 2 and R ⊂ S is seminormal and infra-integral.   (3) In the last case, set   In case (1), S is a local ring. It follows that its maximal ideal is J giving x 2 ∈ M for each x ∈ J. The same holds if R ⊂ S is minimal.
(2) Conversely, let J be an ideal of S such that J R and J [2] ⊆ M. Set T := R + J and let z ∈ T \ R. There exist some a ∈ R, y ∈ J such that z = a + y.   Proof. Since R ⊂ S is quadratic, S t = R[t] for each t ∈ S\R. Moreover, C t := (R : S t ) is a radical ideal of S t and R, and R/C t is Artinian by [7,Lemma 4.8,Theorem 4.2], so that C t = M. From dim k (S t /M) ≤ 2, we deduce that k ⊂ S t /M is minimal, and so is R ⊂ S t . Hence R ⊂ S is a pointwise minimal extension.
If |k| > 2, then, k ⊂ S/M, and R ⊂ S are minimal extensions by Proposition 4.3 (1). If |k| = 2, the proof of Proposition 4.3 (2) shows that k ⊆ k n is quadratic for any integer n > 1.
An integral extension R ⊂ S has FIP as soon as M := (R : S) ∈ Max(R) is such that R/M ⊂ S/M satisfies condition (4) of Theorem 5. 3. The next proposition shows that in many cases, a pointwise minimal FIP integral extension is actually a minimal extension, completing Proposition 5.1. (1) There exists a pointwise minimal extension k ⊂ S which is neither a pointwise minimal pair nor an FCP extension.
For each i ∈ I, let x i be the class of X i in S. Then, S is a zero-dimensional local ring with maximal ideal M := ({x i } i∈I ) and k ⊂ S is subintegral. Let x ∈ M, there exists a finite set J ⊂ I, such that x = i∈J a i x i , with a i ∈ k for each i ∈ J. Then, x 2 = 0, so that k ⊂ k[x] is minimal ramified, and k ⊂ S is pointwise minimal but is not a pointwise minimal pair by Theorem 5.2(1) (2), because x i x j = 0 for i, j ∈ I, i = j. If |I| = ∞, then k ⊂ S has not FCP by [7,Theorem 4.2] and if |I| < ∞, then k ⊂ S has FCP by the same reference. Moreover, if |k| = ∞ and |I| > 2, then k ⊂ S has not FIP by Proposition 6.3 since k is infinite and k ⊂ S is not minimal. Indeed, if k ⊂ S has FIP, there exists some y ∈ S such that S = k[y] ([1, Theorem 3.8], which would imply that k ⊂ S is minimal. If |k| < ∞ and |I| = 2, then k ⊂ S is an FIP pointwise minimal extension which is not a pointwise minimal pair. (2) There exists a pointwise minimal pair k ⊂ T which is neither a co-pointwise minimal extension nor an FCP extension.
For each i ∈ I, let x i be the class of X i in T . Then, T is a zero-dimensional local ring with maximal ideal M := ({x i } i∈I ) and k ⊂ T is subintegral. For x ∈ M, there is a finite set J ⊂ I such that x = i∈J a i x i , with a i ∈ k for each i ∈ J. Then, x 2 = 0, so that k ⊂ k[x] is minimal ramified, and k ⊂ T is a pointwise minimal pair by Theorem 5.2 (2), because x i x j = 0 for i, j ∈ I. If |I| = ∞, then k ⊂ S has not FCP [7,Theorem 4.2]. If |I| < ∞, then k ⊂ S has FCP by the same reference. If |k| = ∞ and |I| > 2, then k ⊂ S has not FIP (same reason as in (1)).
If |k| < ∞ and |I| = 3, then k ⊂ S is an FIP pointwise minimal pair, but is not co-pointwise minimal by Corollary 5.6 There exists a co-pointwise minimal extension which is not an FIP extension. Here, k is an infinite field. Let S := k(X 1 , X 2 ) be the field of rational functions over k, where X 1 , X 2 are two indeterminates and set R := k(X 2 1 , X 2 2 ). Then, R ⊂ S is a height one radicial field extension of degree 4 because [S : In view of Corollary 5.6, R ⊂ S is a co-pointwise minimal extension. But R ⊂ S has not FIP by Proposition 6.3 since k is infinite and R ⊂ S is not minimal, for if not its degree would be 2.
(4) An FIP co-pointwise minimal extension exists by Corollary 5.6(2). (5) We give a last example, showing that condition (1) of Theorem 5.2 with k = t S k = + S k = S may occur. Here k is an infinite field and k ⊂ K is a minimal radicial field extension of degree 2. For any y ∈ K \ k, we have K = k[y] = k + ky, with y 2 ∈ k. Fix such an y and set S := K[X]/(X 2 ), where X is an indeterminate, and let x be the class of X in S. Then, K ⊂ S is minimal ramified by Lemma 2.4, so that S is a local ring with maximal ideal M = Kx = kx + kxy satisfying S/M ∼ = K, and S = k[x, y] = k + kx + ky + kxy. Set T := k + M = k + kx + kxy, so that S = T [y] = T + T y = K + M. It is easy to see that (T, M) is a zero-dimensional local ring. Moreover, k ⊂ T is subintegral and T ⊂ S is inert since k ∼ = T /M ⊂ S/M ∼ = K is a minimal radicial field extension. To end, k ⊂ S/M is a height one radicial extension as well as k ⊂ S, M [2] = 0 and t 2 ∈ k for each t ∈ S (to see this, write t = a + by + x(c + dy), with a, b, c, d ∈ k; then, t 2 = a 2 + b 2 y 2 ∈ k). Hence, k ⊂ S satisfies Theorem 5.2(1). Moreover, since x 2 = 0, we get that k ⊂ k[x] is minimal ramified and k ⊂ k[y] is minimal inert.
Remark 6. 5. We may remark that an extension k ⊂ S, satisfying Theorem 5.2(1), with k = t S k = + S k = S, has not FIP. Indeed, since k ⊂ S/N is radicial, k needs to be an infinite field because any finite extension of a finite field is separable. Then, k ⊂ S has to be minimal in view of Proposition 6.3, a contradiction. Moreover, Example 6.4(5) shows that there exists a pointwise minimal extension k ⊂ S with x, y ∈ S \ k such that k ⊂ k There exists x ∈ T ′ such that T ′ = k[x] is a local ring with maximal ideal N ′ := kx and such that x 2 = 0. Moreover, there exists y ∈ S ′ \ T ′ , such that S ′ = T ′ [y] is a local ring with maximal ideal P = kx + ky, with y 2 , xy ∈ N ′ . Then, S ′ = k + kx + ky and y ∈ P . It follows that y 2 = 0 by Theorem 5.2 (1), and xy ∈ N ′ , which gives that xy = ax, for some a ∈ k. Hence x(y − a) = 0 in S ′ . If a = 0, then y − a is a unit in S ′ , and x = 0, a contradiction. This implies that a = 0 since {1, x, y} is a free system over k and S ′ ∼ = k[X, Y ]/(X 2 , Y 2 , XY ) (there is a surjective k-algebra morphism k[X, Y ]/(X 2 , Y 2 , XY ) → S ′ between two vector spaces whose dimensions are equal).
If R ⊂ T and T ⊂ S are both decomposed and R ⊂ S is pointwise minimal, then k ⊂ T ′ and T ′ ⊂ S ′ are both decomposed, and k ⊂ S ′ is a pointwise minimal FIP extension. It follows that condition (3) of Theorem 5.2 is satisfied. By [7, Lemma 5.4], S ′ has 3 maximal ideals, whose intersection is (0) and S ′ ∼ = k 3 , which gives in fact condition (4) of Theorem 5.2 since |k| = 2.
None of the conditions of [6, Theorem 2.2] holds for a pointwise minimal FIP extension, so that |[R, S]| > 3. This can be easily seen: for each t ∈ S \ T , R ⊂ R[t] is minimal, with R[t] = T, S, R. 6. 3. Transfer properties with respect to Nagata extensions. We are now looking at the transfer properties of pointwise minimal extensions (resp. pairs) with respect to Nagata rings. To get new results, we consider only non-minimal extensions R ⊂ S since R ⊂ S is minimal if and only if R(X) ⊂ S(X) is minimal by [8,Theorem 3.4].
Proposition 6.7. A non-minimal ring extension R ⊂ S with M := (R : S) ∈ Max(R) is a pointwise minimal extension (resp. pair, copointwise minimal extension) if and only if R(X) ⊂ S(X) is a pointwise minimal extension (resp. pair, co-pointwise minimal extension), except for the case where |R/M| = 2 and R ⊂ S is a seminormal infra-integral extension. Proof. Observe that R ⊂ S is integral (resp. integrally closed) if and only if so is R(X) ⊂ S(X) [8, Proposition 3.8]. Since a pointwise minimal extension (resp. pair, co-pointwise minimal) is either integral or integrally closed by Proposition 3.4, it is enough to assume that R ⊂ S is either integral or integrally closed. In the same way, R ⊂ S is minimal if and only if so is R(X) ⊂ S(X) [8,Theorem 3.4]. Since a pointwise minimal integrally closed extension is minimal, we delete this condition. Now, assume that R ⊂ S is integral with M := (R : S) ∈ Max(R). Then MR(X)S(X) ⊆ R(X) and MR(X) ∈ Max(R(X)) give that (R(X) : S(X)) = MR(X). Assume also that we have either |R/M| = 2 or R ⊂ S is not a seminormal infra-integral extension (see Remark 6.8).
We  3.15]. If these conditions hold, let N be the maximal ideal of S, which is also the maximal ideal of U so that NS(X) is the maximal ideal of S(X). Any element of S(X) (resp. NS(X)) is of the form f (X) = P (X)/Q(X), where P (X) ∈ S[X] (resp. NS[X]) and Q(X) ∈ Σ. If R ⊂ S satisfies (C), then N [2] ⊆ M. Let f (X) = P (X)/Q(X) ∈ NS(X), with P (X) = a i X i , where a i ∈ N for each i. Then, P (X) 2 = a 2 i X 2i + 2 a i a j X i+j . From (a i + a j ) 2 = a 2 i + a 2 j + 2a i a j ∈ M, with a 2 i , a 2 j ∈ M, we deduce 2a i a j ∈ M, so that f (X) 2 ∈ MR(X). The converse is obvious.
The properties of being a height one radicial field extension and of being of characteristic p are transmitted to Nagata rings. At last, if c(k) =: p , then N [p] ⊆ k if and only if NT (X) [p] ⊆ k(X), in a similar way as it was proved just before. Then U ⊂ S is a height one radicial extension if and only if so is U(X) ⊂ S(X).
To conclude, R ⊂ S satisfies (C) if and only if R(X) ⊂ S(X) satisfies (C). Then we have the equivalence of pointwise minimal extension (resp. pair) for R ⊂ S and R(X) ⊂ S(X).
At last, in view of Proposition 3.10, R ⊂ S is a co-pointwise minimal extension if and only if R ⊂ S is a pointwise minimal pair such that ℓ[R, S] = 2. Since ℓ[R, S] = ℓ[R(X), S(X)] for an FCP extension R ⊂ S by [19,Theorem 3.3] and R(X) ⊂ S(X) is a co-pointwise minimal extension if and only if R(X) ⊂ S(X) is a pointwise minimal pair such that ℓ[R(X), S(X)] = 2, we get that R ⊂ S is a co-pointwise minimal extension if and only if so is R(X) ⊂ S(X).

Lattices properties of pointwise minimal extensions
We introduce here FMC pairs since we will use them. 7. 1. FMC pairs. An extension R ⊂ S is called an FMC pair if R ⊂ T has FMC for each T ∈ [R, S]. We intend to show that FMC pairs are nothing but FCP extensions. For some results already known, we give shorter proofs.
We temporarily introduce a definition. An extension U ⊆ V is called F MC ⋆ n if there is a finite maximal chain from U to V with length ≤ n and U ⊆ U has FCP (or equivalently, has FMC).
Theorem 7. 1. Let R ⊆ S be a ring extension. The following conditions are equivalent: (1) R ⊆ S has FCP; (2) There exists a finite maximal chain C from R to S with R ∈ C; (3) R ⊂ S is an FMC pair; (4) R ⊂ S and R ⊆ R have FMC. We are going to show by induction on n that R ⊆ S has FCP, under the above assumption. The induction hypothesis is: F MC ⋆ n ⇒ F CP . If n = 1, then R ⊆ S is minimal and has FCP.
Assume that the induction hypothesis holds for n − 1. Let C := {R i } n i=0 be a finite maximal chain with length n such that R 0 = R and R n = S. Then, R ⊂ R 1 is minimal and C 1 := {R i } n i=1 is a finite maximal chain with length n − 1.
If R ⊂ R 1 is minimal integral, then R 1 ⊆ R, so that R is also the integral closure of R 1 ⊆ S. Moreover, R 1 ⊆ R has FCP. The induction hypothesis gives that R 1 ⊆ S has FCP, and so has R ⊆ S. To conclude, R ⊆ S has FCP by [7,Theorem 3.13].
If R ⊂ R 1 is Prüfer minimal, set N := C(R, R 1 ). Then, N ∈ Supp R (R/R) in view of Lemmata 2. 8  We are now going to look at the lattice properties of pointwise minimal extensions or pairs. Before, we give the following lemma. (1) If R ⊆ S has FCP, then [R, S] is a complete Noetherian Artinian lattice for intersection and compositum, whose least element is R and S is its largest element. For lattice properties, we use the definitions and results of [24]. (2) An element T of [R, S] is an atom (resp.; co-atom) if and only if R ⊂ T (resp.; T ⊂ S) is a minimal extension. Now R ⊂ S is called: (a) semimodular if, for each T 1 , T 2 ∈ [R, S] such that T 1 ∩ T 2 ⊂ T i is minimal for i = 1, 2, then T i ⊂ T 1 T 2 is minimal for i = 1, 2.
(b) atomistic if each element of [R, S] is the join (or the least upper bound) of a set of atoms (see [22, page 80]). In fact, this is equivalent to each T ∈ [R, S] is the compositum of the atoms contained in T .
Conversely, assume that R ⊂ S is a finitely geometric FCP extensions, so that [R, S] is semimodular. Let T ∈ [R, S] and x ∈ S\T . Since R ⊂ S has FCP, there exists a maximal finite chain of R-subalgebras of T , R = R 0 ⊂ R 1 ⊂ · · · ⊂ R n−1 ⊂ R n = T , where each R i ⊂ R i+1 is minimal and of course, x ∈ R i . We are going to show by induction on i that R i ⊂ R i [x] is minimal for each i ∈ {0, . . . , n}. The property holds for i = 0 since R ⊂ S is pointwise minimal. Assume that R i ⊂ R i [x] is minimal for some i ∈ {0, . . . , n − 1}. Then, is minimal. Since this property holds for each i, we get that T = R n ⊂ T [x] = R n [x] is minimal, and R ⊂ S is a pointwise minimal pair and an affine pair since R ⊂ S is finitely atomistic. Assume first that R ⊂ S has FCP. Then, R ⊂ S is an affine pair by Lemma 7.3 and there is a finite independent set I of atoms such that S = T I by Proposition 7.5 (1) and [12,Theorem 4,p.174] (take a minimal set I of atoms such that S = T I ).
Conversely, assume that there is a finite independent set I of n atoms with S = T I and that R ⊂ S is an affine pair. Since R ⊂ S is an affine pair and a pointwise minimal pair, [R, S] is semimodular. By [12,Theorem 4,p.174], ℓ[R, S] = n is finite and then R ⊂ S has FCP.