RINGS WHOSE MODULES ARE DIRECT SUMS OF EXTENDING MODULES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, vol.137, no.7, pp.2265-2271, 2009 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 137 Issue: 7
- Publication Date: 2009
- Doi Number: 10.1090/s0002-9939-09-09807-4
- Journal Name: PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Page Numbers: pp.2265-2271
- Open Archive Collection: AVESIS Open Access Collection
- Yıldız Technical University Affiliated: No
Abstract
We prove that for a ring R, the following are equivalent: (i) Every right R-module is a direct sum of extending modules, and (ii) R has finite type and right colocal type (i.e., every indecomposable right R-module has simple socle). Thus, in this case, R is two-sided Artinian and right serial, and every right R-module is a direct sum of finitely generated uniform modules. This property of a ring is not left-right symmetric. A consequence is the following: R is Artinian serial if and only if every R-module is a direct sum of extending modules if and only if R is left serial with every right R-module a direct sum of extending modules.