RINGS WHOSE MODULES ARE DIRECT SUMS OF EXTENDING MODULES


Er N. F.

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, vol.137, no.7, pp.2265-2271, 2009 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 137 Issue: 7
  • Publication Date: 2009
  • Doi Number: 10.1090/s0002-9939-09-09807-4
  • Title of Journal : PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
  • Page Numbers: pp.2265-2271

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.