Back to Search
Start Over
Rings whose cyclics are -modules.
- Source :
- Journal of Algebra & Its Applications; Oct2016, Vol. 15 Issue 8, p-1, 18p
- Publication Year :
- 2016
-
Abstract
- It is well known that if every cyclic right module over a ring is injective, then the ring is semisimple artinian. This classical theorem of Osofsky promoted a considerable interest in the rings whose cyclics satisfy a certain generalized injectivity condition, such as being quasi-injective, continuous, quasi-continuous, or . Here we carry out a study of the rings whose cyclic modules are -modules. The motivation is the observation that a ring is semisimple artinian if and only if every -generated right -module is a -module. Many basic properties are obtained for the rings whose cyclics are -modules, and some structure theorems are proved. For instance, it is proved that a semiperfect ring has all cyclics -modules if and only if it is a direct product of a semisimple artinian ring and finitely many local rings, and that a right self-injective regular ring has all cyclics -modules if and only if it is a direct product of a semisimple artinian ring, a strongly regular ring and a matrix ring over a strongly regular ring. Applications to the rings whose -generated modules are -modules, and the rings whose cyclics are ADS or quasi-continuous are addressed. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 02194988
- Volume :
- 15
- Issue :
- 8
- Database :
- Complementary Index
- Journal :
- Journal of Algebra & Its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 116973402
- Full Text :
- https://doi.org/10.1142/S0219498816501528