Indecomposable Decomposition and couniserial dimension

Dimensions like Gelfand, Krull, Goldie have an intrinsic role in the study of theory of rings and modules. They provide useful technical tools for studying their structure. In this paper we define one of the dimensions called couniserial dimension that measures how close a ring or module is to being uniform. Despite their different objectives, it turns out that there are certain common properties between the couniserial dimension and Krull dimension like each module having such a dimension contains a uniform submodule and has finite uniform dimension, among others. Like all dimensions, this is an ordinal valued invariant. Every module of finite length has couniserial dimension and its value lies between the uniform dimension and the length of the module. Modules with countable couniserial dimension are shown to possess indecomposable decomposition. In particular, von Neumann regular ring with countable couniserial dimension is semisimple artinian. If the maximal right quotient ring of a non-singular ring R has a couniserial dimension as an R-module, then R is a semiprime right Goldie ring. As one of the applications, it follows that all right R-modules have couniserial dimension if and only if R is a semisimple artinian ring.

0. Introduction In this article we introduce a notion of dimension of a module, to be called couniserial dimension. It is an ordinal valued invariant that is in some sense a measure of how far a module is from being uniform. In order to define couniserial dimension for modules over a ring R, we first define, by transfinite induction, classes ζ α of R-modules for all ordinals α ≥ 1. First we remark that if a module M is isomorphic to all its non-zero submodules, then M must be uniform. To start with, let ζ 1 be the class of all uniform modules. Next, consider an ordinal α > 1; if ζ β has been defined for all ordinals β < α, let ζ α be the class of those R-modules M such that for every non-zero submodule N of M, where N ≇ M, we have N ∈ β<α ζ β . If an R-module M belongs to some ζ α , then the least such α is called the couniserial dimension of M, denoted by c.u.dim(M). For M = 0, we define c.u.dim(M) = 0. If a non-zero module M does not belong to any ζ α , then we say that c.u.dim(M) is not defined, or that M has no couniserial dimension. Equivalently, Proposition 2.3 shows that an R-module M has couniserial dimension if and only if for each descending chain of submodules of M, M 1 ≥ M 2 ≥ ..., there exists n ≥ 1, either M n is uniform or M n ∼ = M k for all k ≥ n. It is clear by the definition that every submodule and so every summand of a module with couniserial dimension has couniserial dimension. Also note that, for the integer number n, couniserial dimension of Z n is n. An example is given to show that the direct sum of two modules each with couniserial dimension (even copies of a module) need not have couniserial dimension. In Section 2, we prove some basic properties of the couniserial dimension. In Section 3, we prove our main results. It is shown in Theorem 3.3 that a module of countable (finite or infinite) couniserial dimension can be decomposed in to indecomposable modules. Theorem 3.5 shows that a Dedekind finite module with couniserial dimension is a finite direct sum of indecomposable modules. Theorem 3.10 in Section 3 shows that for a right non-singuar ring R with maximal right quotient ring Q, if Q R has couniserial dimension, then R is a semiprime right Goldie ring which is a finite product of piecewise domains. The reader may compare this with the wellknown result that a prime ring with Krull dimension is a right Goldie ring but need not be a piecewise domain. Furthermore, a prime right Goldie ring need not have couniserial dimension as is also the case for Krull dimension.
In Section 4, we give some applications of couniserial dimension. It is shown in Proposition 4.2 that a module M with finite length is semisimple if and only if for every submodule N of M the right R-module ⊕ ∞ i=1 M/N has couniserial dimension. As a consequence a commutative noetherian ring R is semisimple if and only if for every finite length module M the module ⊕ ∞ i=1 M has couniserial dimension. It is shown in Proposition 4.4 that if P is an anti-coHopfian projective right R-module and ⊕ ∞ i=1 E(P ) has couniserial dimension, then P is injective. As another application we show that all right (left) Rmodule have couniserial dimension if and only if R is semisimple artinian (see Theorem 4.8). Several examples are included in the paper that demonstrates as to why the conditions imposed are necessary and what, if any, there is any relation with the corresponding result in the literature.

. Definitions and Notation.
Recall that a semisimple module M is said to be homogeneous if M is a direct sum of pairwise isomorphic simple submodules. A module M has finite uniform dimension (or finite Goldie rank) if M contains no infinite direct sum of non-zero submodules, or equivalently, there exist independent uniform submodules U 1 , ..., U n in M such that ⊕ n i=1 U i is an essential submodule of M. Note that n is uniquely determined by M. In this case, it is written u.dim(M) = n.
For any module M, we define Z(M) = {x ∈ M : r.ann(x) is an essential right ideal of R} . It can be easily checked that Z(M) is a submodule of M. If Z(M) = 0, then M is called a non-singular module. In particular, if we take A ring R is a called right Goldie ring if it satisfies the following two conditions: (i) R has ascending chain condition on right annihilator ideals and, (ii) u.dim(R R ) is finite.
Recall that a ring R is right V-ring if all right simple R-modules are injective. A ring R is called fully right idempotent if I = I 2 , for every right ideal I. We recall that a right V-ring is fully right idempotent (see [19,Corollary 2.2]) and a prime fully right idempotent ring is right non-singular (see [2,Lemma 4.3]). So a prime right V-ring is right non-singular. Recall that a module M is called Σ-injective if every direct sum of copies of M is injective. A ring R is called right Σ-V-ring if each simple right module is Σ-injective.
In this paper, for a ring R, Q = Q max (R) stands maximal right quotient ring R. It is well known that if R is a right non-singular, then the injective hull of R R , E(R R ), is a ring and is equal to the maximal right quotient ring of R, [9, Corollary 2.31].
A module M is called Hopfian if M is not isomorphic to any of its proper factor modules (equivalently, every onto endomorphism of M is 1-1). Anti-Hopfian modules are introduced by Hirano and Mogami [13]. Such modules are isomorphic to all its non-zero factor modules. A module M is called uniserial if the lattice of submodules are linearly ordered. Anti-Hopfian modules are uniserial artinian.
Recall that a module M is called coHopfian if it is not isomorphic to a proper submodule (equivalently, every 1-1 endomorphism of M is onto). Varadarjan [22] dualized the concept of anti-Hopfian module and called it anti-coHopfian module. With slight modification we will call a non-zero module to be anti-coHopfian if is isomorphic to all its non-zero submodules. A non-zero module M is called uniform if the intersection of any two non-zero submodules is non-zero. We see an anti-coHopfian module is noetherian and uniform.
An R-module M has cancellation property if for every R-modules N and T , M ⊕ N ∼ = M ⊕ T implies N ∼ = T . Every module with semilocal endomorphism ring has cancellation property [16]. Since endomorphism ring of a simple module is a division ring, it has cancellation property.
Throughout this paper, let R denote an arbitrary ring with identity and all modules are assumed to be unitary and right modules, unless other words stated. If N is a submodule (resp. proper submodule) of M we write N ≤ M (resp. N < M). Also, for a module M, ⊕ ∞ i=1 M stands for countably infinite direct sum of copies of M. If N is a submodule of M and k > 1,

. Basic and Preliminary Results.
As defined in the introduction, couniserial dimension is an ordinal valued number. The reader may refer to [21] regarding ordinal numbers. We begin this section with a lemma and a remark on the definition of couniserial dimension.

Lemma 2.1 An anti-coHopfian module is uniform noetherian.
Proof. Since M is isomorphic to each cyclic submodules, M is cyclic and every submodule of M is cyclic and so M is noetherian. Thus M has a uniform submodule, say U. Since U ∼ = M, M is uniform. The next proposition provides a working definition for a module M that has couniserial dimension. As a consequence, we have the following corollary. Proof. The proof is by transfinite induction on c.u.dim(M) = α. The case α = 1 is clear. Let α > 1 and 0 ≤ β < α, then, using Remark 2.2, there exists a submodule K of M such that K ≇ M and β ≤ c.u.dim(K). Now since β ≤ c.u.dim(K) < α, by induction hypothesis, there exists a submodule N of K such that c.u.dim(N) = β.
As a consequence we have the following. Lemma 2.6 Every module with couniserial dimension has a uniform submodule.
In the next proposition we observe that every module of finite couniserial dimension has finite uniform dimension. Example 2.8 There exist modules of infinite couniserial dimension but of finite uniform dimension. Take M = Z p ∞ ⊕ Z p ∞ . Then M is artinian Zmodule of infinite couniserial dimension but of finite uniform dimension 2.
In the following we consider equality in the above lemma in a special case.
This completes the proof.
Note that the condition being injective is necessary in the above proposition.
Example 2. 10 We can see easily that for M = Z⊕Z, c.u.dim(M) = u.dim(M) = 2 but M is not semisimple. Also, the next lemma shows that there exists a module of finite uniform dimension without couniserial dimension.
The following lemma shows that direct sum of two uniform modules may not have couniserial dimension. Proof. Let I be a non-cyclic right ideal of D. Choose a non-zero element x ∈ I. Set J 1 = xR which is isomorphic to D. Thus there exists a right ideal J 2 of D such that J 2 ∼ = I and J 2 ≤ J 1 . Now let J 3 be a cyclic right ideal contained in J 2 and by continuing this manner we have a descending chain J 1 ≥ J 2 ≥ ... of right ideals of D where for each odd integer i, J i is cyclic and for each even integer i, J i is not cyclic. Now consider the descending chain S ⊕ J 1 ≥ S ⊕ J 2 ≥ ... of submodules of S ⊕ D. Since S has cancellation property and for each i, S ⊕ J i is not uniform, by using Proposition 2.3, we see that, for some n, J n ∼ = J n+1 , a contradiction. Thus D is a principal right ideal domain. Remark 2.12 (1) The simple module S in the statement of the Lemma 2.11 can be replaced by any cancellable module. Indeed it follows from the Theorem 3.10, proved latter if the maximal right quotient ring Q of a domain D as Dmodule has couniserial dimension, then Q D has cancellation property and so if Q ⊕ D as D-module has couniserial dimension, D must be right principal ideal domain.
(2) Also, since a Dedekind domain has cancellation property, similar proof shows that if D is a Dedekind domain which is not right principal ideal domain, then D⊕D does not have couniserial dimension. This example shows that even direct sum of a uniform module with itself may not have couniserial dimension.
We call an R-module M fully coHopfian if every submodule of M is coHopfian. Note that artinian modules are fully coHopfian. If I is the set of prime numbers, then ⊕ p∈I Z p is an example of fully coHopfian Z-module that it is not artinian.
If α is a limit ordinal and 1 ≤ β < α, then by Remark 2.2, there exists a non-zero submodule K of M 2 such that β ≤ c.u.dim(K). Then by induction The condition fully coHopfian of Proposition 2.14 is necessary. Let us recall the definition of uniserial dimension [20].

Definition 2.17
In order to define uniserial dimension for modules over a ring R, we first define, by transfinite induction, classes ζ α of R-modules for all ordinals α ≥ 1. To start with, let ζ 1 be the class of non-zero uniserial modules. Next, consider an ordinal α > 1; if ζ β has been defined for all ordinals β < α, let ζ α be the class of those R-modules M such that, for every submodule If M is nonzero and M does not belong to any ζ α , then we say that "u.s.dim(M) is not defined," or that " M has no uniserial dimension." Using the above remark we have the following interesting results.
.. = M n = M and for each i > n, M i = 0. Then, by Proposition 2.3, there exists n ≥ 1 such that This shows M is injective.

. Main Results.
In this section we use our basic results to prove the main results. Remark 3.2 One can see that the above result provides another proof for the fact that commutative V-rings (i.e, von Neumann regular rings ) are Σ-V-ring. For an example of right V-rings that is not Σ-V-ring, the reader may refer to [15,Example,page 60].
The next result shows that if a module has countable couniserial dimension then it can be decomposed into indecomposable modules.
Remark 3. 4 We do not know whether the above proposition holds for a module of arbitrary couniserial dimension. For infinite countable couniserial dimension one can show under some condition that the module can be represented as a direct sum of uniform modules.
Recall that a module M is called Dedekind finite if M is not isomorphic to any proper direct summand of itself. Clearly, every direct summand of a Dedekind finite module is a Dedekind finite module. Obviously, a Hopfian module is Dedekind finite. Since all finitely generated modules over a commutative ring are Hopfian (see [10]), they provide examples of Dedekind finite modules. A ring R is called a von Neumann regular ring if for each x ∈ R, there exists y ∈ R such that xyx = x, equivalently, every principal right ideal is a direct summand. R is unit regular ring if for each x ∈ R, there exists a unit element u ∈ R such that x = xux. As a consequence of the above theorem we have the following corollary.

Corollary 3.6 Every Dedekind finite von Neumann regular ring (in particular, unit regular rings ) with couniserial dimension is semisimple artinian.
A ring R is called a PWD (piecewise domain) if it possesses a complete set {e i |0 ≤ i ≤ n} of orthogonal idempotents such that xy = 0 implies x = 0 or y = 0 whenever x ∈ e i Re k and y ∈ e k Re j . Note that the definition is left-right symmetric and all e i Re i are domain, see [12].
An element x of R is called regular if its right and left annihilators are zero.
Proposition 3.7 Let R be a semiprime right Goldie ring with couniserial dimension. If u.dim(R R ) = n, then R has a decomposition into n uniform modules. In particular, it is a piecewise domain.
Proof. We can assume that n > 1. Let I 1 = U 1 ⊕ ... ⊕ U n be an essential right ideal of R. Then, by [8, Proposition 6.13], I 1 contains a regular element x and thus J 1 = xR is a right ideal of R which is R-isomorphic to R. So u.dim(J 1 ) = n and it contains an essential right ideal I 2 of R such that it is a direct sum of n uniform right ideals. By continuing in this manner we obtain a descending chain I 1 ≥ J 1 ≥ I 2 ≥ ... of right ideals of R such that I i are direct sum of n uniform and J i are isomorphic to R. Since R has couniserial dimension, for some n, I n ∼ = R. The last statement follows from [12,. This completes the proof.  Proof. Let M ≥ M 1 ≥ M 2 ≥ ... be a descending chain of Q-submodules of M. So it is a descending chain of R-submodules of M and thus, for some n, M n is uniform R-module or M n ∼ = M i as R-modules for all i ≥ n. If M n is uniform R-module, then it is also uniform Q-module. So let M n ∼ = M i as R-modules and let ϕ i be this isomorphism. If q ∈ Q and t ∈ M n there exists an essential right ideal E of R such that qE ≤ R. So ϕ i (tqE) = ϕ i (tq)E and also ϕ i (tqE) = ϕ i (t)qE. Then ϕ i (tq)E = ϕ i (t)qE. Since Q is right non-singular, ϕ i (tq) = ϕ i (t)q. Thus ϕ i is a Q-isomorphism. This completes the proof.  Proof. It is enough to show that R has finite uniform dimension. Since Q R has couniserial dimension, R R has couniserial dimension and so every right ideal of R has couniserial dimension. Thus Lemma 2.6 implies that every right ideal contains a uniform submodule. Now by [8,Theorem 3.29] the maximal right quotient ring of R is a product of right full linear rings, say Q = i∈I Q i , where Q i are right full linear rings. Note that since R R is right non-singular, Q R is also non-singular and so, using Lemma 3.9, Q Q has couniserial dimension. At first we claim each Q i is endomorphism ring of a finite dimensional vector space. Assume the contrary. Then Q j is the endomorphism ring of an infinite dimensional vector space, for some j. Thus Q j ∼ = Q j ×Q j and so if ι : Q j −→ Q be the canonical embedding, then ι(Q j ) is a right ideal of Q and there exists a Q-isomorphism Q ∼ = ι(Q j ) × Q. Then there exist right ideals T 1 and T of Q such that Q = T 1 ⊕ T , T 1 and Q are isomorphic as Q-modules and T ∼ = ι(Q j ) as Q-module. Because Q j is the endomorphism ring of an infinite dimensional vector space, it has a right ideal which is not principal, for example its socle. So ι(Q j ) and thus T contains a non-cyclic right ideal of Q and thus since T ∼ = Q/T 1 , there exists a non-cyclic right ideal of Q, say K 1 such that Q ≥ K 1 ≥ T 1 . Now T 1 is isomorphic to Q. So we can have a descending chain Q > K 1 > T 1 > K 2 > T 2 > ... of right ideals of Q such that T i are cyclic but K i are not cyclic. This is a contradiction. So all Q i are endomorphism ring of finite dimensional vector spaces. Now to show R is semiprime right Goldie ring it is enough to show that the index set I is finite. If I is infinite, there exist infinite subsets I 1 and I 2 of I such that I = I 1 ∪ I 2 . and I 1 ∩ I 2 is empty. Let T 1 = i∈I N i such that N i = Q i for all i ∈ I 1 and N i = 0 for all i ∈ I 2 . Similarly let T = i∈I M i such that M i = Q i for all i ∈ I 2 and M i = 0 for all i ∈ I 1 . Then T 1 and T are right ideals of Q and Q = T 1 ⊕ T . T contains a right ideal of Q which is not cyclic, for example ⊕ i∈I M i . Since T ∼ = Q/T 1 , there exists a non-cyclic right ideal K 1 of Q such that Q ≥ K 1 ≥ T 1 . Note that T 1 is a cyclic Q-module and because I 1 is infinite, the structure of T 1 is similar to that of Q. We can continue in this manner and find a descending chain of right ideals of Q such that K i are non cyclic Q-modules and T i are cyclic Q modules, which is a contradiction. Therefore I is finite and R must have finite uniform dimension. This shows R is semiprime right Goldie ring and so Proposition 3.7 and [12, Corollary 3] imply that it is a direct sum of prime right Goldie rings.
The reader may ask what if R R has couniserial dimension instead of Q R . Indeed we may point out that unlike a semiprime ring with right Krull dimension, a semiprime ring with couniserial dimension need not be a right Goldie ring. See Dubrovin [5] that contains an example of a primitive uniserial ring with non-zero nilpotent elements.
Next we show that the converse of the above theorem is not true, in general. In fact we show that there exists a prime right Goldie ring R such that c.u.dim(R R ) = 2 and Q R does not have couniserial dimension. We need the following lemma to give the example. Example 3.12 Here we give an example of a prime right Goldie ring R with maximal right quotient ring Q such that Q R does not have couniserial dimension. Take R = M 2 (Z), the 2 × 2 matrix ring over Z. Then R is a prime right Goldie ring with maximal right quotient ring Q = M 2 (Q). Note that under the standard Morita equivalent between the ring Z and R = M 2 (Z), see [17,Theorem 17.20 ], R corresponds to Z ⊕ Z and so using the above lemma R has couniserial dimension 2. If {p i |i ≥ 1} is the set of all prime numbers, then Q/Z = ∞ i=1 K i /Z, where K i = {m/p n i |n ≥ 0 and m ∈ Z}. Then take Q n = ∞ i=n K i . Then M 2 (Q 1 ) ≥ M 2 (Q 2 ) ≥ ... is a descending chain of R-submodules of Q which are not uniform R-modules. Assume that for some n, M 2 (Q n ) ∼ = M 2 (Q n+1 ) with an R-isomorphism φ. Let where p n does not odd non of t i,j for all 1 ≤ i ≤ 4. Then since φ is additive, we can easily see that m 1,j p j n /t 1,j m 2,j p j n /t 2,j m 3,j p j n /t 3,j m 4,j p j n /t 4,j = m 1 /t 1 m 2 /t 2 m 3 /t 3 m 4 /t 4 and this implies that p j n |m i for all j ≥ 1 and 0 ≤ i ≤ 4 and so m i = 0, a contradiction. So Q R does not have couniserial dimension.    Recall that a ring R is called right bounded if every essential right ideal contains a two-sided ideal which is essential as a right ideal. A ring R is called right fully bunded if every prime factor ring is right bounded. A right noetherian right fully bounded ring is commonly abbreviated as a right FBN ring. Clearly all commutative noetherian rings are example of right FBN rings. Finite matrix rings over commutative noetherian rings are a large class of right FBN rings which are not commutative. In [ Proof. We first show that P has cancellation property. Let M = P ⊕B ∼ = P ⊕ B ′ . So there exist submodules P ′ and C of M such that M = P ⊕ B = P ′ ⊕ C and P ′ ∼ = P and C ∼ = B ′ . If p 1 is a projection map from M = P ⊕ B on to P . Then with restriction of p 1 to C we have an exact sequence 0 −→ C ∩ B −→ C −→ I −→ 0, such that I is a submodule of P . Note that every submodule of P is projective, because it is anti-coHopfian. So I is projective and thus C ∼ = C ∩ B ⊕ I. Similarly by considering map p 2 from M = P ′ ⊕ C to P ′ we have B ∼ = C ∩ B ⊕ J for some submodule J of P ′ . Since J ∼ = I ∼ = P , we have B ∼ = C and so B ∼ = B ′ . Then P has cancellation property. Now consider the descending chain

. Some Applications.
. Then, by Proposition 2.3, there exists n ≥ 1 such that , because P is cancelable . Since P is finitely generated, there exists a right module L such that for some k, E(P ) k ∼ = P ⊕ L. This shows P is injective.
As a consequence of the above proposition we have the following corollary: Corollary 4.5 Let R be a principal right ideal domain with maximal right quotient ring Q ( which is a division ring). If the right R-module ⊕ ∞ i=1 Q has couniserial dimension, then R = Q.
We need the following lemmas to prove the next theorem. Using Proposition 2.3 we can see that:  Proof. For equivalence of (1), (4) and (5)  (2) ⇒ (1). At first we show R satisfies ascending chain condition on two sided deals. Let I 1 ≤ I 2 ≤ ... be a chain of ideals of R. Since the right module ⊕ ∞ i=1 R/I i has couniserial dimension, there exists n such that, for each j ≥ n, ⊕ ∞ i=n R/I i ∼ = ⊕ ∞ i=j R/I i . Thus they have the same annihilators and so for each j ≥ n, I n = I j . Suppose R is a non-semisimple ring. By Lemma 4.6 every module over a factor ring of R also has couniserial dimension. Thus by invoking the ascending chain condition on two sided ideals we may assume R is not semisimple artinian but every factor ring of R is semisimple artinian. Using Lemma 2.20, R is a right V-ring. First let us assume that R is primitive. So, by Theorem 3.10, R is a prime right Goldie ring. By [4,Theorem 5.16], a prime right V-ring right Goldie is simple. By Lemma 4.7, R has a right noetheian uniform submodule and so using [8, Corollary 7.25], R is right noetherian. Now we show that R is Morita equivalent to a domain. By [7, lemma 5.12], the endomorphism ring of every uniform right ideal of a prime right Goldie ring is a right ore domain. So by [11,Theorem 1.2], it is enough to show that R has a uniform projective generator U. Let us assume that R is not uniform and u.dim(R) = n and let U be a uniform right ideal of R. By [8,Corollary 7.25], U n can be embedded in R and also R can be embedded in U n . Then c.u.dim(R) = c.u.dim(U n ) and hence R ∼ = U n , because R is not uniform. Thus U is a projective generator uniform right ideal of R. So R is Morita equivalent to a domain. Now Lemma 3.11 and Lemma 2.11 and Corollary 4.5 show that R is simple artinian, a contradiction. So R is not primitive, but every primitive factor ring is artinian (indeed all proper factor rings are artinian). Then since R is a right V-ring, by [3], R is regular and Σ-V-ring. Also every right ideal contains a non-zero uniform right ideal, hence minimal. So R has non-zero essential soc(R). But R is Σ-V-ring and by Corollary 2.19 , we have only finitely many non-isomorphic simple modules. Thus soc(R) is injective. This implies R is semisimple, a contradiction. This completes the proof.

Summary
This paper defines couniserial dimension of a module that measures how far a module is from being uniform. The results proved in the paper demonstrate its importance for studding the structure of modules and rings and is a beginning of a larger project to study its impact. We close with some open questions: 1) Does a module with arbitrary couniserial dimension possesses indecomposable dimension? 2) Is there a theory for modules with both finite uniserial and couniserial dimensions that parallels to Krull-Schmidt-Remak-Azumaya theorem?