Semiperfect Modules with Respect to a Preradical

• Published in: Communications in algebra Vol. 34; no. 3; pp. 841 - 856
• Main Authors: Özcan, A. Çiğdem, Alkan, Mustafa
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 01.02.2006
• Subjects: Harada and co-Harada module; Noetherian QF-module; Projective cover; Projective module; Semiperfect module; Semisimple module
• ISSN: 0092-7872; 1532-4125
• DOI: 10.1080/00927870500441593

In this article, we consider the module theoretic version of I-semiperfect rings R for an ideal I which are defined by Yousif and Zhou ( 2002 ). Let M be a left module over a ring R, N ∈ σ[M], and τ M a preradical on σ[M]. We call N τ M -semiperfect in σ[M] if for any submodule K of N, there exists a decomposition K = A ⊕ B such that A is a projective summand of N in σ[M] and B ≤ τ M (N). We investigate conditions equivalent to being a τ M -semiperfect module, focusing on certain preradicals such as Z M , Soc, and δ M . Results are applied to characterize Noetherian QF-modules (with Rad(M) ≤ Soc(M)) and semisimple modules. Among others, we prove that if every R-module M is Soc-semiperfect, then R is a Harada and a co-Harada ring.