On the Loewy Length and the Noetherian Dimension of Artinian Modules

Karamzadeh, O. A. S.; Nejad, A. R. Sajedi

doi:10.1080/00927870209342372

Cited by 14 publications

(9 citation statements)

References 12 publications

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“…It is well known that Artinian modules over Max-rings are Noetherian; see [18]. These facts and Remark 21, together with the next result, imply that modules with Krull dimension over a right ue-ring are both Artinian and Noetherian.…”

Section: Let M Be a Non-semisimple Artinian R-module With Compositionsupporting

confidence: 54%

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On rings with a unique proper essential right ideal

2004

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Abstract. Right ue-rings (rings with the property of the title, i.e., with the maximality of the right socle) are investigated. It is shown that a semiprime ring R is a right ue-ring if and only if R is a regular V-ring with the socle being a maximal right ideal, and if and only if the intrinsic topology of R is non-discrete Hausdorff and dense proper right ideals are semisimple. It is proved that if R is a right self-injective right ue-ring (local right ue-ring), then R is never semiprime and is Artin semisimple modulo its Jacobson radical (R has a unique non-zero left ideal). We observe that modules with Krull dimension over right ue-rings are both Artinian and Noetherian. Every local right ue-ring contains a duo subring which is again a local ue-ring. Some basic properties of right ue-rings and several important examples of these rings are given. Finally, it is observed that rings such as C(X), semiprime right Goldie rings, and some other well known rings are never ue-rings.

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Section: Let M Be a Non-semisimple Artinian R-module With Compositionsupporting

confidence: 54%

“…In [18], it is shown that every module with Krull dimension which is also a Loewy module is Artinian. It is well known that Artinian modules over Max-rings are Noetherian; see [18].…”

Section: Let M Be a Non-semisimple Artinian R-module With Compositionmentioning

confidence: 99%

On rings with a unique proper essential right ideal

2004

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“…As for examples of α-quasi short modules, one can similarly use the facts that there are Artinian modules with Noetherian dimension equals to α, see [15]. In view of Proposition 3.1.…”

Section: α-Quasi Short Modules and α-Almost Quasi Noetherian Modulesmentioning

confidence: 99%

“…The dual Krull dimension in [7,8,11,12,13,14,15,16,17] is called Noetherian dimension and in [5] is called N-dimension. This dimension is called Krull dimension in [21].…”

Section: Introductionmentioning

confidence: 99%

On Α -QUASI SHORT MODULES

Davoudian¹

2017

International Electronic Journal of Algebra

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“…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][8][9][10][11][12][13]. In this article, all rings are associative with 1 ¤ 0, and all modules are unital right modules.…”

Section: Introductionmentioning

confidence: 99%

On α-almost Artinian modules

2016

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In this article we introduce and study the concept of˛-almost Artinian modules. We show that each -almost Artinian module M is almost Artinian (i.e., every proper homomorphic image of M is Artinian), wherę 2 f0; 1g. Using this concept we extend some of the basic results of almost Artinian modules to˛-almost Artinian modules. Moreover we introduce and study the concept of˛-Krull modules. We observe that if M is an˛-Krull module then the Krull dimension of M is either˛or˛C 1.

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On the Loewy Length and the Noetherian Dimension of Artinian Modules

Cited by 14 publications

References 12 publications

On rings with a unique proper essential right ideal

On rings with a unique proper essential right ideal

On Α -QUASI SHORT MODULES

On α-almost Artinian modules

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