On rings with restricted minimum condition

Huynh, Dinh Van; Dan, Phan

doi:10.1007/bf01194021

Cited by 17 publications

(9 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…of a module M) assuming certain classes of modules (associated to M) posses certain properties and viceversa . The results in the present paper are of a similar nature and are an outcome of results proved in [1], [2], [3], [4], [5] ; [6], [8] and [9] . In [1] among other results A. W .…”

Section: Introductionsupporting

confidence: 87%

On certain classes of modules

Varadarajan¹

1992

Publicacions Matemàtiques

View full text Add to dashboard Cite

Dedicated to the me7nory of Pere Menal Let X be any class of R-modules containing 0 and closed under isornorpllic images. With any such X we associate three classes FX, FX and ¿5X. 7 .'lie study of some of the closure properties of three classes allows Lis to obtain characterization of Artinian modules dualizing results of Chatters. The theory of Dual Goldie dimension as developed by the author in some of his earlier work plays a crucial role in the present paper.

show abstract

Section: Introductionsupporting

confidence: 87%

On certain classes of modules

Varadarajan¹

1992

Publicacions Matemàtiques

View full text Add to dashboard Cite

show abstract

“…Now, to show that X itself has finite uniform dimension, we need to show that Soc(X ) is finitely generated, or, equivalently, that Soc(X ) has finite composition length. For this purpose, we use a technique developed in [20]. Assume on the contrary that Soc(X ) is infinitely generated, i.e.,…”

Section: Theorem 12 Let M Be a Finitely Generated Right R-module Ifmentioning

confidence: 99%

Some results on V-rings and weakly V-rings

Dinh¹,

Holston²,

Huynh³

2013

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

a b s t r a c tA ring R is called a right weakly V-ring (briefly, a right WV-ring) if every simple right R-module is X -injective, where X is any cyclic right R-module with X R R R . In this note, we study the structure of right WV-rings R and show that, if R is not a right V-ring, then R has exactly three distinct ideals, 0 ⊂ J ⊂ R, where J is a nilpotent minimal right ideal of R such that R/J is a simple right V-domain. In this case, if we assume additionally that R J is finitely generated, then R is left Artinian and right uniserial with composition length 2. We also show that a strictly right WV-ring with Jacobson radical J is a Frobenius local ring if and only if the injective hull of J R is uniserial. Some other results are obtained in the connection with the Noetherian property of right WV-rings and related rings.

show abstract

“…This class has been useful, among other things, for providing some general characterizations of modules which are direct sums of a module from the class A and a module from some other class. In order to point out previous interest in dealing with such direct sum decompositions, we mention the studies of Chatters [5] on rings such that every cyclic module is the direct sum of a projective module and a module of Krull dimension at most an ordinal α, Huynh and Dan [12] on rings such that every cyclic module is a direct sum of a projective module and an Artinian module, and Al-Khazzi and Smith [1] are direct sums of a semisimple and a Noetherian module. A certain class of type dA includes the extending modules, which are defined as modules with the property that every submodule is essential in a direct summand or, equivalently, every closed submodule is a direct summand [10].…”

Section: Introductionmentioning

confidence: 99%