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Volume 14, Issue 1 (Jan 2016)

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On α-almost Artinian modules

Maryam Davoudian
  • Corresponding author
  • Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran (Islamic Republic of)
  • Email:
/ Ahmad Halali
  • Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran (Islamic Republic of)
  • Email:
/ Nasrin Shirali
  • Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran (Islamic Republic of)
  • Email:
Published Online: 2016-06-24 | DOI: https://doi.org/10.1515/math-2016-0036

Abstract

In this article we introduce and study the concept of α-almost Artinian modules. We show that each α-almost Artinian module M is almost Artinian (i.e., every proper homomorphic image of M is Artinian), where α ∈ {0,1}. Using this concept we extend some of the basic results of almost Artinian modules to α-almost Artinian modules. Moreover we introduce and study the concept of α-Krull modules. We observe that if M is an α-Krull module then the Krull dimension of M is either α or α + 1.

Keywords: Krull dimension; α-Krull module; α-almost Noetherian module; Noetherian dimension; α-short module

MSC 2010: 16P60; 16P20; 16P40

1 Introduction

The concept of Noetherian dimension of a module M, (the dual of Krull dimension of M, in the sense of Rentschler and Gabriel, see [1, 2]) introduced in Lemonnier [3], and Karamzadeh [4], is almost as old as Krull dimension of M, and their existence are equivalent. Later, Chambless [5] studied dual Krull dimension and called it N—dimension. Roberts [6] calls this dual dimension again Krull dimension. The latter dimension is also called dual Krull dimension in some other articles, see for example, [7, 8]. The former dimension has recently received some attention; see [713]. In this article, all rings are associative with 1 ≠ 0, and all modules are unital right modules. If M is an R-module, then n-dim M and k-dim M will denote the Noetherian dimension and the Krull dimension of M. Let us give a brief outline of this paper. In section 2, we introduce and study the concept of α-almost Artinian modules.

Hein [14] introduced almost Artinian modules and studied some of their properties. We say that a module M is α-almost Artinian if for each nonzero submodule N of M, kdimMN<α and α is the least ordinal number with this property. Using this concept, we observe that each α-almost Artinian module M has Krull dimension and k-dim M ≤ α. Consequently, if M is an almost Artinian module, then either M is Artinian or k-dim M = 1. By applying the previous facts we prove more general results and obtain most of the results in [14], as a consequence of these general results. Section 3 is devoted to the concept of α-Krull modules, which is the dual of α-short modules, see [15]. We obtain the dual of each single result in [15], except [15, Proposition 2.1], whose dual is, in fact, not true for α-Krull modules. In the last section we also investigate some properties of α-almost Artinian and α-Krull modules. Finally, we should emphasize here that the results in section 2 are new and are the dual of the corresponding results in [10, 15] and at the same time are the extensions of the results in [14]. The results in sections 3 and 4, are also new, which are the dual of the corresponding results in [15] (we should admit here that some of the proofs in Section 4 can be easily imitated from the proofs of our corresponding results in [15], but we present them for completion). If a nonzero R-module M has Krull dimension and α is an ordinal number, then M is called α-critical if k-dim M = α and kdimMN<α for all nonzero submodules N of M. An R-module M is called critical if M is α-critical for some ordinal α. For all concepts and basic properties of rings and modules which are not defined in this paper, we refer the reader to [1, 12, 16].

We recall that an R-module M is called an almost Artinian module if MN is Artinian for each nonzero submodule N of M, see [14]. It is trivial to see that every almost Artinian R-module is either Artinian or 1-critical.

2 α-almost Artinian modules

In this section we introduce and study α-almost Artinian modules. We extend some of the basic results of almost Artinian modules to α-almost Artinian modules.

We begin with our definition of α-almost Artinian modules.

Definition 2.1: An R-module M is called α-almost Artinian, if for each nonzero submodule N of M, kdimMN<α and α is the least ordinal number with this property.We should remind the reader that the above concept is in fact the dual of α-almost Noetherian modules, see [15, Definition 1.6]. Clearly each α-almost Artinian module M, where α ∊ {0,1}, is almost Artinian (note, in fact if α = 0 then M is simple, i.e., it is 0-critical and if α = 1, then it is either Artinian or 1-critical). We thus consider the condition of a module being α-almost Artinian as a generalization of the condition of a module being almost Artinian.

Remark 2.2: If M is an α-almost Artinian module, then each submodule and each factor module of M is β-almost Artinian for some βα.The next three trivial, but useful facts, which are the dual of the corresponding facts in [15, Lemmas 1.7, 1.8, 1.9] are needed.

Lemma 2.3: If M is an α-almost Artinian module, then M has Krull dimension and k-dim Mα. In particular, k-dim M = α if and only if M is α-critical.

Lemma 2.4: If M is a module with k-dim M = α, then either M is α-critical, in which case it is α-almost Artinian, or it is α + 1-almost Artinian.

Lemma 2.5: If M is an α-almost Artinian module, then either M is α-critical or α = k-dim M + 1. In particular, if M is an α-almost Artinian module, where α is a limit ordinal, then M is α-critical.

The following is now immediate.

Corollary 2.6: Let M be a β + 1-almost Artinian module, then either k-dim M = β or k-dim M = β + 1.

Proposition 2.7: An R-module M has Krull dimension if and only if M is α-almost Artinian for some ordinal α.

Corollary 2.8: Every α-almost Artinian module has finite uniform dimension.

By Corollary 2.8, every α-almost Artinian module admits finite indecomposable direct decompositions. The next proposition provides criteria for an α-almost Artinian module to be indecomposable.

Proposition 2.9: Let M be an α-almost Artinian module. Then M is indecomposable if either α is a limit ordinal or k-dim M = α.

The following corollary is now immediate.

Corollary 2.10: If M is an α-almost Artinian module, then either M is indecomposable or k-dim M = β, where α = β + 1.

The following lemma which is the dual of [10, Proposition 2.2] and the next few results are needed for our study in this article.

Lemma 2.11: If R is a commutative ring and M is an α-critical module, then for each rR we have either AnnM(r) = 0 or AnnM(r) = M.

The following result is now immediate.

Lemma 2.12: Let R be a commutative ring and M be an α-critical R-module, then AnnR(M) is a prime ideal of R.

The following corollary, being a trivial consequence of the previous fact, is a generalization of [14, Theorem 1.1, c].

Corollary 2.13: Let R be a commutative ring and M be an α-almost Artinian module. If αk-dim M + 1, then AnnR(M) is a prime ideal of R.

Lemma 2.14: Let R be a commutative ring and M be an α-critical R-module, then M is a torsion-free α-critical RAnnR(M)-module.

In view of the previous lemma and Lemma 2.5, the following corollary is now immediate.

Corollary 2.15: Let R be a commutative ring and M be an α-almost Artinian module. If αk-dim M + 1, then M is a torsion-free α-critical RAnnR(M)-module.

We also have the following lemma about critical modules.

Lemma 2.16: Let R be a commutative ring. If M is an α-critical R-module, then M is isomorphic to a submodule of the quotient field of RAnnR(M).

In view of the previous lemma and Lemma 2.5, the following corollary, which is a generalisation of [14, Theorem 1.1, f], is now immediate.

Corollary 2.17: Let R be a commutative ring and M be an α-almost Artinian module. If α ≠ k-dim M + 1, then M is isomorphic to a submodule of the quotient field of RAnnR(M).

In the case of critical modules we have the following proposition.

Proposition 2.18: If M is an α-critical R-module, then EndR (M) has no nonzero zero divisors.

In view of the previous proposition and Lemma 2.5, we have the following immediate corollary which is a generalization of [14, Theorem 1.1, g].

Theorem 2.19: Let M be an α-almost Artinian module. If αk-dim M + 1, then EndR(M) has no nonzero zero divisors.A commutative ring R is called α-almost Artinian ring, for some ordinal number a, if for every non-zero ideal I of R, kdimRI<α and α is the least ordinal number with this property. We now have the following theorem.

Theorem 2.20: Let R be a commutative ring. If R is β+1-almost Artinian, then either k-dim R = β or R is a β+1-critical domain.

The next theorem is a generalization of [14, Theorem 1.3].

Theorem 2.21: Let M be an α-critical R-module, where R is a commutative ring. Then there exists a prime ideal P such that RP is an α-critical domain. In particular, if M contains a torsion-free element (i.e., there exists x ∊ M such that ann.(x) = 0), then R itself is an α-critical domain.

The next result is dual of [10, Corollary 2.4].

Proposition 2.22: If M is an R-module, then the following are equivalent:

  • (1)

    M is critical.

  • (2)

    Every nonzero submodule of M is essential in M and M has a critical submodule N with kdimMN<kdimN.

Next, we show that there is a long chain of submodules of a Noetherian module M, whose factor modules are critical of the same dimension α, where α is any ordinal less than k-dim M.

Proposition 2.23: If M is a Noetherian module which is not Artinian, that is k-dim M ≥ 1. Then for each ordinal α < k-dim M there exists an infinite chain of submodules MM1M2 ⊃ … in M such that MiMi+1 is α-critical for each i.

3 α-Krull modules

We begin with the following definition, which is in fact the dual of α-short modules, see [15, Definition 1.1], and in the subsequent results we try to present counterparts of the appropriate results in [15].

Definition 3.1: An R-Module Mis called α-Krull, if for each submodule N of M, either k-dim N ≤ α or kdimMNα, and α is the least ordinal number with this property.

Remark 3.2: If M is an R-module with k-dim M = α, then M is β-Krull for some β ≤ α.

Remark 3.3: If M is an α-Krull module, then each submodule and each factor module of M is β-Krull for some β ≤ α.

The proof of the following lemma is similar to the proof of its dual in [15, Lemma 1.4] and is therefore omitted.

Lemma 3.4: If M is an R-module and for each submodule N of M, either MN has Krull dimension, then so does M.

The previous result and Remark 3.2, immediately yield the next result.

Corollary 3.5: Let M be an α-Krull module. Then M has Krull dimension and k-dim M ≥ α.

Proposition 3.6: An R-module M has Krull dimension if and only if M is α-Krull for some ordinal α.

Corollary 3.7: Every α-Krull module has finite uniform dimension.

Proposition 3.8: If M is an α-Krull R-module, then either k-dim M = α or k-dim M = α = 1.

Corollary 3.9: If M is a 0-Krull module, then either k-dim M = 1 or M is Artinian.

In view of Proposition 3.8, the following remark is now evident.

Remark 3.10: If M is a β-Krull R-module, then it is an α-almost Artinian module for some β ≤ α ≤ β + 2. We claim that all the cases in the latter inequality can occur. To see this, we note that every 1-critical module is 0-Krull which is also 1-almost Artinian and every α-critical module, where α is a limit ordinal, is an α-Krull module which is also α-almost Artinian (note, for every ordinal α, there exists an α-critical module, see the comment at the end of this section). Finally, there exists a 2-almost Artinian module which is 0-Krull, see Example 4.9.

Remark 3.11: An R-module M is —1-Krull if and only if it is simple. Thus any —1-Krull module is 0-conotable and 0-critical (note, an R-module M is called α-conotable, if n-dim M = α and n-dim N < α for each proper submodules N of M).

Proposition 3.12: Let M be an R-module, with k-dim M = α, where α is a limit ordinal. Then M is α-Krull.

Proposition 3.13: Let M be an R-module and k-dim M = α = β + 1. Then M is either α-Krull or it is β-Krull.

For critical modules we have the following proposition.

Proposition 3.14: Let M be an α-critical R-module, where α = β + 1. Then M is a β-Krull module.

The following remark, which is a trivial consequence of the previous fact, shows that the converse of Proposition 3.12, is not true in general.

Remark 3.15: Let M be an α + 1-critical R-module, where α is a limit ordinal. Then M is an α-Krull module.

Proposition 3.16: Let M be an R-module such that k-dim M = α + 1. Then M is either an α-Krull R-module or there exists a submodule N of M such that kdimMN=kdimN=α+1.

Proposition 3.17: Let M be a nonzero α-Krull R-module. Then either M is β-almost Artinian for some ordinal β ≤ α + 1 or there exists a submodule N of M with k-dim N ≤ α.

Finally, we conclude this section by providing some examples of α-almost Artinian (resp. α-Krull) modules, where a is any ordinal.

First, we recall that if M is a Noetherian R-module with k-dim M = α, then for any ordinal β ≤ a there exists a β-critical R-submodule of M, see the comment which follows [12, Proposition 1.11]. We also recall that given any ordinal α there exists a Noetherian module M such that k-dim M = α, see [11, Example 1], and [17]. Consequently, we may take M to be a Noetherian module with k-dim M = α and for any ordinal β < α, we take N to be its β-critical submodule, then by Lemma 2.4, N is β-almost Artinian module. We recall that the only α-almost Artinian modules, where α is a limit ordinal, are α-critical modules, see Lemma 2.5. Therefore to see an example of an α-almost Artinian module which is not α-critical, the ordinal α must be a non-limit ordinal. Thus we may take M to be a non-critical module with k-dim M = β, where α = β + 1, see [11, Example 1], hence it follows trivially that M is an α-almost Artinian module. As for examples of α-Krull modules, one can similarly use the facts that there are Noetherian modules M with Krull dimension equal to α and for each β ≤ α there are β-critical submodules of M and then apply Propositions 3.12, 3.13, 3.14, to give various examples of α-Krull modules (for example, by Proposition 3.14, every α + 1-critical module is α-Krull).

4 Properties of α-Krull modules and α-almost Artinian modules

In this section some properties of α-Krull modules, α-almost Artinian modules over an arbitrary ring R are investigated.

Lemma 4.1: Let M be an R-module. If there exists a submodule K of M such that k-dim K ≤ α and MN is an α-Krull module. Then M is α-Krull.

The previous lemma has the following analogue, whose proof is similar, but we give it for completeness.

Lemma 4.2: Let M be an R-module. If there exists a submodule K of M such that K is an α-Krull R-module and kdimMKα. Then M is α-Krull.

Corollary 4.3: Let R be a ring and M be an R-module. If M = M1 ⊕ M2 such that M1 is an α-Krull module and k-dim M2 < α, then M is α-Krull.We note that the module P ∞ is Artinian and the -module ℤ is a 0-Krull module. By the previous corollary, P ⊕ ℤ is a 0-Krull module. It is also clear that P ⊕ ℤ is not Artinian.

Proposition 4.4: Let M be an R-module. If M contains submodules L ⊆ N such that NL is α-Krull, kdimMNα, and k-dim L ≤ α, then M is α-Krull.

The next two results are now in order.

Proposition 4.5: Let R be a ring and M be a nonzero α-Krull module, which is not a critical module, then M contains a submodule L such that k-dim L ≤ α.

Proposition 4.6: Let M be an R-module. If there exists a submodule N of M such that N is α-Krull, MN is β-Krull, and μ = sup{α, β}, then M is γ-Krull such that μ ≤ γ ≤ μ + 1.

Using Lemma 2.5, we give the next immediate result which is the counterpart of the previous proposition for α-almost Artinian modules.

Proposition 4.7: Let M be an R-module. If there exists a submodule N of M such that N is α-almost Artinian, MN is β-almost Artinian, and μ = sup{α, β}, then M is γ-almost Artinian such that μ ≤ γ ≤ μ + 1.

Corollary 4.8: Let R be a ring. If M1 is an α1-Krull (resp. α1-almost Artinian) R-module and M2 is an α2-Krull (resp. α2- almost Artinian) R-module and let α = sup{α1, α2}. Then M1 ⊕ M2 is μ-Krull (resp. μ- almost Artinian) for some ordinal number μ such that α ≤ μ ≤ α + 1.

The next example shows that in the previous corollary we may have all the cases for μ.

Example 4.9: If M1 = M2 = p, then M1 andM2 are 0-Krull (resp. 1-almost Artinian) ℤ-modules such that M1 ⊕ M1 is also 0-Krull (resp. 1-almost Artinian). Now let M1 = M2 = ℤ. In this case the ℤ-module ℤ is 0-Krull (resp. 1 -almost Artinian), but the ℤ-module ℤ ⊕ ℤ is 1 -Krull (resp. 2-almost Artinian). Finally ℤp 0 ℤ is a 0-Krull ℤ-module which is 2-almost Artinian.

Theorem 4.10: Let M be a non-zero R-module. Let α be an ordinal number. Suppose that for every proper factor K of M there exists an ordinal number γ ≤ α such that K is γ-Krull. In that case either k-dim M = α + 1 or M is μ-Krullfor some ordinal number μ ≤ a. In particular, M is μ-Krullfor some ordinal μ ≤ α + 1.

The next result is the dual of Theorem 4.10.

Theorem 4.11: Let a be an ordinal number and M be an R-module such that every proper submodule of M is γ-Krull for some ordinal number γ ≤ α. If α = — 1, then M is also μ-Krull for some μ ≤ 0. If not, then M is μ-Krull where μ ≤ α. Moreover, k-dim M ≤ α + 1.

Corollary 4.12: Let M be a module. If every proper submodule of M is 0-Krull, then so is M.

Remark 4.13: If every nonzero proper submodule of an R-module M is —1-Krull, then every nonzero proper submodule of M is both a maximal and a minimal submodule of M, and vice versa.

The following example shows that in the previous theorem we may have μ = α + 1.

Example 4.14: Let M = A ⊕ B, where A and B are simple R-modules. Clearly M is 0-Krull. We know that every nonzero proper submodule P of M is simple (i.e., P is —1-Krull).

The next immediate result is the counterparts of Theorems 4.10, 4.11, for α-almost Artinian modules.

Proposition 4.15: Let M be an R-module and α be an ordinal number. If each proper submodule N of M (resp. each proper factor module of M) is γ- almost Artinian with γ < α, then M is a μ-almost Artinian module with #x03BC; ≤; α + 1, k-dim M ≤ α (resp. with μ ≤; α + 1, k-dim M ≤; α + 1).

By Lemma 2.3 (Corollary 3.5) every α-almost Artinian (resp. α-Krull) module has Krull dimension and thus by [18, Corollary 6] has Noetherian dimension. Consequently, we have the following immediate result.

Proposition 4.16: The following statements are equivalent for a ring R.

  • Every R-module with Krull dimension is Noetherian.

  • Every α-Krull R-module is Noetherian for all α.

  • Every α-almost Artinian R-module is Noetherian for all α.

Moreover, if R is a right perfect ring (i.e., every R-module is a Loewy module) then every α-Krull resp. α-almost Artinian) R-module is both Artinian and Noetherian, see [11, Proposition 2.1].

Before concluding this section with our last observation, let us cite the next result which is in [11, Theorem 2.9], see also [13, Theorem 3.2].

Theorem 4.17: For a commutative ring R the following statements are equivalent.

  • Every R-module with finite Noetherian dimension is Noetherian.

  • Every Artinian R-module is Noetherian.

  • Every R-module with Noetherian dimension is both Artinian and Noetherian.

Now in view of the above theorem and the well-known fact that each domain with Krull dimension 1 is Noetherian, see [1, Proposition 6.1] and also [12, Corollary 2.15], we observe the following result.

Proposition 4.18: The following statements are equivalent for a commutative ring R.

  • Every Artinian R-module is Noetherian.

  • Every m-Krull module is both Artinian and Noetherian for all integers m — —1.

  • Every α-Krull module is both Artinian and Noetherian for all ordinals α.

  • Every m-almost Artinian R-module is both Artinian and Noetherian for all non-negative integers m.

  • Every α-almost Artinian R-module is both Artinian and Noetherian for all ordinals a.

  • No homomorphic image of R can be isomorphic to a dense subring of a complete local domain of Krull dimension 1.

Acknowledgement

We would like to thank professor O.A.S. Karamzadeh for helpful discussions on the topics of this paper and for his valuable comments on the preparation of this paper. We are also grateful to an anonymous, meticulous referee for some useful suggestions and finding an error in the earlier version of this article.

References

  • [1]

    Gordon R., Robson J.C., Krull dimension, Mem. Amer. Math. Soc., 1973, 133

  • [2]

    Krause G., On fully left bounded left Noetherian rings, J. Algebra, 1972, 23, 88–99

  • [3]

    Lemonnier B., Deviation des ensembless et groupes totalement ordonnes, Bull. Sci. Math., 1972, 96, 289-303

  • [4]

    Karamzadeh O.A.S., Noetherian-dimension, Ph.D. thesis, Exeter University, England, UK, 1974

  • [5]

    Chambless L., N-Dimension and N-critical modules. Application to Artinian modules, Comm. Algebra, 1980, 8, 1561–1592

  • [6]

    Roberts R.N., Krull-dimension for Artinian Modules over quasi local commutative Rings, Quart. J. Math. Oxford, 1975, 26, 269-273

  • [7]

    Albu T., Vamos P., Global Krull dimension and Global dual Krull dimension of valuation Rings, Lecture Notes in Pure and Applied Mathematics, 1998, 201, 37–54

  • [8]

    Albu T., Smith P.F., Dual Krull dimension and duality, Rocky Mountain J. Math., 1999, 29, 1153–1164

  • [9]

    Karamzadeh O.A.S., Motamedi M., On α-DICC modules, Comm. Algebra, 1994, 22, 1933–1944

  • [10]

    Karamzadeh O.A.S., Sajedinejad A.R., Atomic modules, Comm. Algebra, 2001, 29(7), 2757–2773

  • [11]

    Karamzadeh O.A.S., Sajedinejad A.R., On the Loewy length and the Noetherian dimension of Artinian modules, Comm. Algebra, 2002, 30, 1077–1084

  • [12]

    Karamzadeh O.A.S., Shirali N., On the countability of Noetherian dimension of Modules, Comm. Algebra, 2004, 32, 4073–4083

  • [13]

    Hashemi J., Karamzadeh O.A.S., Shirali N., Rings over which the Krull dimension and the Noetherian dimension of all modules coincide, Comm. Algebra, 2009, 37, 650–662

  • [14]

    Hein J., Almost Artinian modules, Math. Scand., 1979, 45, 198–204

  • [15]

    Davoudian M., Karamzadeh O.A.S., Shirali N., On α-short modules, Math. Scand., 2014, 114 (1), 26-37

  • [16]

    Anderson F.W., Fuller K.R., Rings and categories of modules, Springer-Verlag, 1992

  • [17]

    Fuchs L., Torsion preradical and Ascending Loewy series of modules, J. Reine und Angew. Math., 1969, 239, 169–179

  • [18]

    Lemonnier B., Dimension de Krull et codeviation, Application au theorem d’Eakin, Comm. Algebra, 1978, 6, 1647-1665

  • [19]

    Bilhan G., Smith P.F., Short modules and almost Noetherian modules, Math. Scand., 2006, 98, 12–18

  • [20]

    Albu T., Smith P.F., Localization of modular lattices, Krull dimension, and the Hopkins-Levitzki Theorem(I), Math. Proc. Cambridge Philos. Soc., 1996, 120, 87–101

  • [21]

    Albu T., Smith P.F., Localization of modular lattices, Krull dimension, and the Hopkins-Levitzki Theorem (II), Comm. Algebra, 1997, 25, 1111–1128

  • [22]

    Albu T., Teply L., Generalized deviation of posets and modular lattices, Discrete Math., 2000, 214, 1–19

  • [23]

    Karamzadeh O.A.S., Motamedi M., a-Noetherian and Artinian modules, Comm. Algebra, 1995, 23, 3685–3703

  • [24]

    Kirby D., Dimension and length for Artinian modules, Quart. J. Math. Oxford, 1990, 41, 419–429

  • [25]

    McConell J.C., Robson J.C., Noncommutative Noetherian Rings, Wiley-Interscience, New York, 1987

About the article

Received: 2015-01-11

Accepted: 2016-04-14

Published Online: 2016-06-24

Published in Print: 2016-01-01



Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0036. Export Citation

© 2016 Davoudian et al., published by De Gruyter Open.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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