#### Title

Some Results on V-rings and Weakly V-rings

#### Publication Title

Journal of Pure and Applied Algebra

#### Publication Date

1-2013

#### Document Type

Article

#### DOI

10.1016/J.JPAA.2012.04.001

#### Keywords

16D50, 16D60, 16D70, 16U10, ring theory, artin rings, modules (algebra), nilpotent groups, jacobson radical

#### Disciplines

Algebra | Applied Mathematics | Mathematics

#### Abstract

A 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*. In this note, we study the structure of right WV-rings

_{R}*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

*is finitely generated, then*

_{R}J*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*is uniserial. Some other results are obtained in the connection with the Noetherian property of right WV-rings and related rings.

_{R}#### Recommended Citation

Dinh, Hai Q.; Holston, Christopher J.; and Huynh, Dinh V. (2013). Some Results on V-rings and Weakly V-rings. *Journal of Pure and Applied Algebra* 217(1), 125-131. doi: 10.1016/J.JPAA.2012.04.001 Retrieved from https://digitalcommons.kent.edu/mathpubs/14

#### Publisher

Elsevier

#### Publisher Location

Amsterdam, Netherlands