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 ∈ M , and M a preradical on M . We call N M -semiperfect in 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 M and B ≤ M N . We investigate conditions equivalent to being a M -semiperfect module, focusing on certain preradicals such as Z M Soc, and 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.

A generalization of projective covers

“…Since S ⊆ δ(P ), we have that S = δ(S). Since S is projective, S is semisimple by [12,Proposition 2.12]. Then there exists a summand Y of P such that Y ⊆ S and P = Y ⊕ X.…”

“…A module M is said to have a projective cover (δ-cover [15], Soc-cover [12], respectively) if there exists an epimorphism f : P → M such that P is projective and Ker f P (Ker f δ P , Ker f ⊆ Soc(P ), respectively). Here we consider some generalizations of these covers.…”

“…Recently some authors have worked with various extensions of these rings (see for example [1,9,11,12,14,15]). Zhou calls a ring R δ-semiperfect (δ-semiregular) if for a (finitely generated) left ideal I of R, I = Rf ⊕ S where f 2 = f ∈ R and S ⊆ δ( R R), and also by defining projective δ-cover, he characterizes these rings.…”

“…Next, Nicholson and Zhou study I -semiperfect and I -semiregular rings by introducing strongly lifting ideals. A module theoretic version of these extensions is studied in [1,12]; also by defining projective Soc-covers, Soc-semiperfect modules are characterized in [12].…”

Modules in σ[M] with Chain Conditions on δ_M-small Submodules

“…We write them for convenience Ozcan and Alkan, (2006) Recall that a module M is said to have a uniform dimension n, where n is a nonnegative integer ,if n is the maximal number of summands in a direct sum of nonzero submodules of M. In this case we write u.dim M = n and we say M has a finite uniform dimension. is not finitely cogenerated} is nonempty .…”