Semiperfect Modules with Respect to a Preradical

ABSTRACT

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.

Key Words:

• Harada and co-Harada module
• Noetherian QF-module
• Projective module
• Projective cover
• Semiperfect module
• Semisimple module

Mathematics Subject Classification:

• 16D40
• 16D50
• 16L60
• 16D99

ACKNOWLEDGMENTS

Mustafa Alkan was supported by the Scientific Research Project Administration of Akdeniz University. We are deeply grateful to Prof. Dr. R. Wisbauer for encouraging us to consider our results in due generality and for various valuable comments. Also we would like to thank the referee for all helpful suggestions and careful reading of the previous version of the manuscript.

Notes

Communicated by R. Wisbauer.