RAD-SUPPLEMENTING MODULES
JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY, vol.53, no.2, pp.403-414, 2016 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 53 Issue: 2
- Publication Date: 2016
- Doi Number: 10.4134/jkms.2016.53.2.403
- Journal Name: JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Page Numbers: pp.403-414
- Keywords: supplement, Rad-supplement, supplementing module, Rad-supplementing module, perfect ring
- Open Archive Collection: AVESIS Open Access Collection
- Dokuz Eylül University Affiliated: Yes
Abstract
Let R be a ring, and let M be a left R-module. If M is Rad-supplementing, then every direct summand of M is Rad-supplementing, but not each factor module of M. Any finite direct sum of Rad-supplementing modules is Rad-supplementing. Every module with composition series is (Rad-)supplementing. M has a Rad-supplement in its injective envelope if and only if M has a Rad-supplement in every essential extension. R is left perfect if and only if R is semilocal, reduced and the free left R-module (R-R)((N)) is Rad-supplementing if and only if R is reduced and the free left R-module (R-R)((N)) is ample Rad-supplementing. M is ample Rad-supplementing if and only if every submodule of M is Rad-supplementing. Every left R-module is (ample) Rad-supplementing if and only if R/P(R) is left perfect, where P(R) is the sum of all left ideals I of R such that Rad I = I.