Startseite Mathematik On α-almost Artinian modules
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On α-almost Artinian modules

  • Maryam Davoudian EMAIL logo , Ahmad Halali und Nasrin Shirali
Veröffentlicht/Copyright: 24. Juni 2016
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 14 Heft 1

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 [7–13]. 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, k−dim⁡MN<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 k−dim⁡MN<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-moduleMis called α-almost Artinian, if for each nonzero submoduleNofM, k−dim⁡MN<αandαis 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

IfMis an α-almost Artinian module, then each submodule and each factor module ofMis β-almost Artinian for someβ ≤ α.

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

IfMis an α-almost Artinian module, thenMhas Krull dimension andk-dim M ≤ α. In particular, k-dim M = αif and only ifMis α-critical.

Lemma 2.4

IfMis a module withk-dim M = α, then eitherMis α-critical, in which case it is α-almost Artinian, or it isα + 1-almost Artinian.

Lemma 2.5

IfMis an α-almost Artinian module, then eitherMis α-critical or α = k-dim M + 1. In particular, ifMis an α-almost Artinian module, whereαis a limit ordinal, thenMis α-critical.

The following is now immediate.

Corollary 2.6

LetMbe aβ + 1-almost Artinian module, then eitherk-dim M = βork-dim M = β + 1.

Proposition 2.7

An R-moduleMhas Krull dimension if and only ifMis 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

LetMbe an α-almost Artinian module. ThenMis indecomposable if eitherαis a limit ordinal ork-dim M = α.

Proof

It suffices to observe that M is critical (note, an α-critical module is indecomposable, see [1, Corollary 2.5 and Proposition 2.6]). We have two cases:

  1. α is a limit ordinal : By Lemma 2.5, M is an α-critical R-module.

  2. α is not a limit ordinal: Based on the hypothesis, k-dim M = α and so M is critical by Lemma 2.3. □

The following corollary is now immediate.

Corollary 2.10

IfMis an α-almost Artinian module, then eitherMis indecomposable ork-dim M = β, whereα = 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

IfRis a commutative ring andMis an α-critical module, then for eachr ∊ Rwe have either AnnM(r) = 0 or AnnM(r) = M.

Proof

We know that MAnnM(r)≃rM. If AnnM(r) ≠ 0, then k−dimrM=k−dimMAnnM(r)<k−dimM, but if rM ≠ 0, then rM is α-critical and α=k−dim⁡rM=k−dimMAnnM(r)<α, which is impossible. Thus rM = 0 i.e., AnnM(r) = M and we are done. □

The following result is now immediate.

Lemma 2.12

Let R be a commutative ring andMbe 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

LetRbe a commutative ring andMbe an α-almost Artinian module. Ifα ≠ k-dim M + 1, thenAnnR(M) is a prime ideal of R.

Proof

By Lemma 2.5, M is α-critical and we are through. □

Lemma 2.14

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

Proof

Clearly M is an α-critical RAnnR(M)-module. Since AnnR(M) is a prime ideal of R we may assume that R is an integral domain and AnnR(M) = 0. Now suppose that r ∊ R and m ∊ M are such that rm = 0 and m ≠ 0. Then AnnM(r) ≠ 0 and by Lemma 2.11, AnnM(r) = M. Therefore r ∊ AnnR(M) = 0 and it follows that M is torsion-free. □

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

Corollary 2.15

LetRbe a commutative ring andMbe an α-almost Artinian module. Ifα ≠ k-dim M + 1, thenMis a torsion-free α-criticalRAnnR(M)-module.

We also have the following lemma about critical modules.

Lemma 2.16

Let R be a commutative ring. IfMis an α-critical R-module, thenMis isomorphic to a submodule of the quotient field ofRAnnR(M).

Proof

Since each non-zero submodule of M is indecomposable, it follows that no two non-zero submodules of M can have a zero intersection. Therefore M is a rank one torsion-free RAnnR(M)-module and it follows that M is isomorphic to a submodule of 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 andMbe an α-almost Artinian module. If α ≠ k-dim M + 1, thenMis isomorphic to a submodule of the quotient field ofRAnnR(M).

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

Proposition 2.18

IfMis an α-critical R-module, then EndR(M)has no nonzero zero divisors.

Proof

Let 0 ≠ f ∊ EndR(M). In that case Mker(f)≃Im(f). If ker(f) ≠ 0, then α = k-dim I m(f) = k−dimMker(f)<k−dim⁡M=α, which is a contradiction. Thus k e r(f) = 0 and f is monomorphism. Now let g be an endomorphism of M such that fg = 0. In that case f(g(M)) = 0 and g(M) = 0 i.e., g = 0. □

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

LetMbe an α-almost Artinian module. If α ≠ k-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, k−dimRI<α and α is the least ordinal number with this property. We now have the following theorem.

Theorem 2.20

LetR be a commutative ring. If R is β+1-almost Artinian, then eitherk-dim R = β or R is a β+1-critical domain.

Proof

By Lemmas 2.3 and 2.4, we have k-dim R = β or R is β+1-critical. Let Rr be a β + 1-critical module. It is sufficient to show that R is a domain. If 0 ≠ r ∊ R, then rR ≠ 0 and it follows that AnnR(r) = 0, see Lemma 2.11. Therefore R is a 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.

Proof

Put ann(M) = Q, then by Lemma 2.12, Q is a prime ideal. Now by considering M as a faithful RQ-module, without lose of generality we may assume that M is a faithful R-module (i.e., ann(M) = 0). If we put ann(x) = Px for each 0 ≠ x ∊ M, then since x R is an α-critical R-module (note, every submodule of an α-critical module is an α-critical module, see [1, Proposition 2.3]), we infer that Px is a prime ideal by Lemma 2.12. We have xR≃RPx, hence RPx is an α-critical domain and we are done. The last part is now evident because we have Px = 0 for some x ∊ M. □

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

Proposition 2.22

IfMis an R-module, then the following are equivalent:

  1. Mis critical.

  2. Every nonzero submodule ofMis essential inMandMhas a critical submoduleNwithk−dimMN<k−dimN.

Proof

(1)⇒(2) It is clear.

(2) ⇒ (1) We may assume that N ≠ M. Thus k−dim⁡M=sup{k−dim⁡N,k−dimMN}=k−dim⁡N. We suppose that there exists a nonzero submodule P of M such that k−dimMP=k−dim⁡M and get a contradiction. We know that k−dimMP=k−dim⁡M/P∩NP/P∩N. Thereforek−dimMP∩N≥k−dim⁡MP=k−dim⁡M. This shows that k−dim⁡M=k−dim⁡N=k−dimMP∩N (note, by our hypothesis N is an essential submodule of M, hence we have 0 ≠ N ∩ P ⊆ P). Therefore k−dimMP∩N=k−dim⁡N>k−dimMN. Since N is critical we infer that k−dimMP∩N=k−dim⁡N>k−dimMP∩N. Consequently, we have k−dimMN=k−dim⁡M/P∩NN/P∩N=k−dimMP∩N, which is a contradiction. □

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

IfMis a Noetherian module which is not Artinian, that isk-dim M ≥ 1. Then for each ordinal α < k-dim Mthere exists an infinite chain of submodulesM ⊃ M1 ⊃ M2 ⊃ … inMsuch thatMiMi+1is α-critical for each i.

Proof

Let M1 be a maximal submodule with respect to the property that k-dim M ≥ α. Clearly for each 0≠M′M1⊆MM1,k−dimMM′<α, hence k−dimMM1≤sup{k−dimM/M1M′/M1,0≠M′M1⊂MM1}+≤α. This shows that k−dimMM1=α and MM1 is an α-critical module. Since k−dimM=sup{k−dimMM1=α,k−dimM1}>α, we infer that k-dim M1 > α. Now by the same argument M1 has a submodule M2 such that M1M2 is α-critical. We repeat this process and get the desired infinite chain. □

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-dimN ≤ α or k−dimMN≤α, and α is the least ordinal number with this property.

Remark 3.2

IfMis an R-module withk-dim M = α, thenMis β-Krull for some β ≤ α.

Remark 3.3

IfMis an α-Krull module, then each submodule and each factor module ofMis 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

IfMis an R-module and for each submoduleNof M, eitherMNhas Krull dimension, then so doesM.

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

Corollary 3.5

LetMbe an α-Krull module. ThenMhas Krull dimension andk-dim M ≥ α.

Proposition 3.6

An R-moduleMhas Krull dimension if and only ifMis α-Krull for some ordinal α.

Corollary 3.7

Every α-Krull module has finite uniform dimension.

Proposition 3.8

IfMis an α-Krull R-module, then eitherk-dim M = α ork-dim M = α = 1.

Proof

In view of Corollary 3.5, we have k-dim M ≥ α. If k-dim M ≠ α, then k-dim M ≥ α + 1. Now let M1 ⊇ M2 ⊇ … be any descending chain of submodules of M. If there exists some k such that k-dim Mk ≤ α, then k−dimMiMi+1≤k−dimMi≤k−dimMk≤α for each i ≥ k. Otherwise k−dimMMi≤α (note, M is α-Krull) for each i, hence k−dimMiMi+1≤α for each i. Thus in any case there exists an integer k such that for each i ≤ k, k−dimMiMi+1≤α. This shows that k-dim M ≤ α + 1, i.e., k-dim M = α + 1. □

Corollary 3.9

IfMis a 0-Krull module, then eitherk-dim M = 1 orMis Artinian.

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

Remark 3.10

IfMis 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-moduleMis —1-Krull if and only if it is simple. Thus any —1-Krull module is 0-conotable and 0-critical (note, an R-moduleMis called α-conotable, ifn-dim M = α andn-dim N < α for each proper submodulesNof M).

Proposition 3.12

LetMbe an R-module, withk-dim M = α, where α is a limit ordinal. ThenMis α-Krull.

Proof

We know that M is β-Krull for some β ≤ α. If β < α, then by Proposition 3.8, k-dim M ≤ β + 1 < α, which is a contradiction. Thus M is α-Krull. □

Proposition 3.13

LetMbe an R-module andk-dim M = α = β + 1. ThenMis either α-Krull or it is β-Krull.

Proof

We know that M is γ-Krull for some γ ≤ α. If γ < β then by Proposition 3.8, we have k-dim M ≤ γ + 1 < β + 1, which is impossible. Hence we are done. □

For critical modules we have the following proposition.

Proposition 3.14

LetMbe an α-critical R-module, where α = β + 1. ThenMis a β-Krull module.

Proof

Let N ⊊ M. In this case k−dimMN<α. Thus k−dimMN<β. This shows that for some βʹ ≤ β, M is βʹ-Krull. If βʹ < β, then βʹ + 1 ≤ β ≤ α. But k-dim M ≤ βʹ + 1 ≤ β ≤ α, by Proposition 3.8, which is a contradiction. Thus βʹ = β and we are done. □

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

LetMbe an α + 1-critical R-module, where α is a limit ordinal. ThenMis an α-Krull module.

Proposition 3.16

LetMbe an R-module such thatk-dim M = α + 1. ThenMis either an α-Krull R-module or there exists a submoduleNofMsuch thatk−dimMN=k−dim⁡N=α+1.

Proof

We know that M is α-Krull or an α + 1-Krull R-module, by Proposition 3.13. Let us assume that M is not an α-Krull R-module, hence there exists a submodule N of M such that k-dim N ≥ α + 1 and k−dimMN≥α+1. This shows that k-dim N = α + 1 and k−dimMN=α+1 and we are through. □

Proposition 3.17

LetMbe a nonzero α-Krull R-module. Then eitherMis β-almost Artinian for some ordinal β ≤ α + 1 or there exists a submoduleNofMwithk-dim N ≤ α.

Proof

Suppose that M is not β-almost Artinian for any β ≤ α + 1. This means that there must exist a submodule N of M such that k−dimMN⪇α. Inasmuch as M is α-Krull, we infer that k-dim N ≤ a and we are done. □

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 thatk-dim K ≤ α andMNis an α-Krull module. Then M is α-Krull.

Proof

Let N be a submodule of M, then k-dim N ∩ K ≤ α. If k−dimNN∩K≤α, then k-dim N ≤ α. Now suppose that k−dimNN∩K>α, then N+KK is a submodule of the α-Krull module MK such that k−dimN+KK>α. Therefore we must have k−dimM/KN+K/K=k−dimMN+K≤α. But k−dimN+KN=k−dimKN∩K≤k−dim⁡K≤α, hence k−dimMN=sup{k−dimN+KN,k−dimMN+K}≤α. This implies that M is β-Krull for some β ≤ α. But MK is α-Krull, hence by Remark 3.3, we must also have α ≤ β and we are done. □

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

Lemma 4.2

LetMbe an R-module. If there exists a submodule K ofMsuch that K is an α-Krull R-module andk−dimMK≤α. ThenMis α-Krull.

Proof

Let N be any submodule of M. Then k−dimN+KK≤k−dimMK≤α. Hence k−dimNN∩K≤α. If k-dim N ∩ K ≤ α, then k-dim N ≤ α. Now suppose that k-dim N ∩ K > α. Since K is α-Krull, we infer that k−dimKK∩N≤α and hence k−dimMN∩K=sup{k−dimKN∩K,k−dimMK}≤α. But

k−dimMN∩K=sup{k−dimKN∩K,k−dimMK}≤α

Therefore k−dimMN≤α. This shows that M is β-Krull for some β ≤ α. But K is α-Krull, hence β ≮ α, i.e., β = α and we are done. □

Corollary 4.3

Let R be a ring andMbe an R-module. IfM = M1 ⊕ M2such that M1is an α-Krull module andk-dim M2 < α, thenMis α-Krull.

We note that the module ℤP ∞ is Artinian and the ℤ-module ℤ is a 0-Krull module. By the previous corollary, ℤP∞ ⊕ ℤ is a 0-Krull module. It is also clear that ℤP∞ ⊕ ℤ is not Artinian.

Proposition 4.4

LetMbe an R-module. IfMcontains submodules L ⊆ Nsuch thatNLis α-Krull, k−dimMN≤α, andk-dim L ≤ α, thenMis α-Krull.

Proof

Since NL is α-Krull and k-dim L ≤ α, then N is α-Krull, by Lemma 4.1. But k−dimMN≤α and since N is α-Krull, M is α-Krull, by Lemma 4.2. □

The next two results are now in order.

Proposition 4.5

Let R be a ring andMbe a nonzero α-Krull module, which is not a critical module, thenMcontains a submodule L such thatk-dim L ≤ α.

Proof

Since M is not critical, we infer that there exists a submodule L ⊊ M, such that k−dimML=k−dim⁡M. We know that k-dim M = α or k-dim M = α + 1, by Proposition 3.8. If k-dim M = α it is clear that k-dimL ≤ α. Hence we may suppose that k−dimML=k−dim⁡M=α+1. Consequently, k-dim L ≤ α and we are done. □

Proposition 4.6

LetMbe an R-module. If there exists a submoduleNofMsuch thatN is α-Krull, MN is β-Krull, and μ = sup{α, β}, thenM is γ-Krull such that μ ≤ γ ≤ μ + 1.

Proof

Since N is α-Krull, thus by Proposition 3.8, k-dim N = α or k-dim N = α + 1. Similarly since MN is β-Krull, k−dimMN=β or k−dimMN=β+1. We infer that M has Krull dimension and k−dim⁡M=sup{k−dimN,k−dimMN}. Therefore μ ≤ k-dim M ≤ μ + 1. But by Remark 3.2, M is γ-Krull for some ordinal number γ and by Proposition 3.8, γ ≤ k-dim M ≤ γ + 1. This shows that γ = μ, or γ = μ + 1 (note, we always have μ ≤ γ) and we are done. □

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

LetMbe an R-module. If there exists a submoduleNofMsuch thatNis α-almost Artinian, MN is β-almost Artinian, and μ = sup{α, β}, thenMis 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 ⊕ M2is 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 = ℤp∞, then M1 andM2 are 0-Krull (resp. 1-almost Artinian) ℤ-modules such that M1 ⊕ M1is 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

LetMbe a non-zero R-module. Let α be an ordinal number. Suppose that for every proper factor K ofMthere exists an ordinal number γ ≤ α such that K is γ-Krull. In that case eitherk-dim M = α + 1 orMis μ-Krullfor some ordinal number μ ≤ a. In particular, Mis μ-Krullfor some ordinal μ ≤ α + 1.

Proof

Let 0 ≠ N⊂M. Since MN is γ-Krull for some ordinal number γ ≤ α, we infer that k−dimMN≤γ+1≤α+1, by Proposition 3.8. This immediately implies that k-dim M ≤ α + 2, see [10, Proposition 1.4]. If k-dim M ≤ α + 1 then we are through. Hence we may suppose that k-dim M = α + 2 and M is not μ-Krull for any μ ≤ α and seek a contradiction. Since M is not μ-Krull for any μ ≤ α, we infer that there must exist a submodule N of M such that k−dimMN≥α+1. But we have already observed that k−dimMN≤α+1, hence k−dimMN=α+1. We now claim that k-dim N ≤ α +1 which trivially implies that k-dim M = α +1 and this is the contradiction that we were looking for. To see this, we note that for any nonZero submodule P of N we must have k−dimNP≤α, for MP is γ-Krull for some γ ≤ α and k−dimM/PN/P=k−dimMN=α+1. But k−dim⁡N≤sup{k−dimNP:0≠P⊆N}+1≤α+1, see [10, Proposition 1.4] and we are done. The final part is now evident. □

The next result is the dual of Theorem 4.10.

Theorem 4.11

Let a be an ordinal number andMbe an R-module such that every proper submodule ofMis γ-Krull for some ordinal number γ ≤ α. If α = — 1, thenMis also μ-Krull for some μ ≤ 0. If not, thenMis μ-Krull where μ ≤ α. Moreover, k-dim M ≤ α + 1.

Proof

If α = —1, then each proper nonℤero submodule of M is both a maximal and a simple submodule of M, i.e. k-dim M = 0. Hence let us assume that α ≥ 0. Now let N ⊊ M be any submodule such that N is γ-Krull for some ordinal number γ with γ ≤ α. We infer that k-dim N ≤ γ + 1 ≤ α + 1, by Proposition 3.8. But we know that k-dim M = sup{k-dim N : N ⊊ M}, see ([10, Proposition 1.4]). This shows that k-dim M ≤ α + 1. If k-dim M ≤ α, then it is clear that M is μ-Krull for some #x03BC; ≤; α. Hence we may suppose that k-dim M = α + 1. If 0 ≠ N ⊊ M is a submodule of M, then we are to show that either that k−dimMN≤α or k-dimN ≤ α. To this end, let us suppose that k-dim N = α + 1 and show that k−dimMN≤α. Now let 0 ≠ N ⊊ N′ ⊊ M. Since N′ is γ-Krull for some γ ≤ α, and k-dim N = α + 1, we must have k−dim⁡N′N≤α. But k−dim⁡MN=sup{k−dimN′N:N′N⊊MN}≤α and we are through. The final part has already been proved. □

Corollary 4.12

LetMbe a module. If every proper submodule ofMis 0-Krull, then so isM.

Remark 4.13

If every nonzero proper submodule of an R-moduleMis —1-Krull, then every nonzero proper submodule ofMis both a maximal and a minimal submodule ofM, and vice versa.

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

Example 4.14

LetM = A ⊕ B, where A and B are simple R-modules. ClearlyMis 0-Krull. We know that every nonzero proper submodule P ofMis 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

LetMbe an R-module and α be an ordinal number. If each proper submoduleNofM(resp. each proper factor module of M) isγ- almost Artinian with γ < α, thenMis 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.

  1. Every R-module with Krull dimension is Noetherian.

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

  3. 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.

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

  2. Every Artinian R-module is Noetherian.

  3. 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.

  1. Every Artinian R-module is Noetherian.

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

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

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

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

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

Proof

Only the proof of (5) → (6) → (1), which is an easy consequence of [19, Proposition 1.3], is needed. □

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.

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Received: 2015-1-11
Accepted: 2016-4-14
Published Online: 2016-6-24
Published in Print: 2016-1-1

© 2016 Davoudian et al., published by De Gruyter Open.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/math-2016-0036
Schlagwörter für diesen Artikel
Krull dimension; α-Krull module; α-almost Noetherian module; Noetherian dimension; α-short module
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Heruntergeladen am 9.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/math-2016-0036/html
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