CS-Rickart modules

Abstract

In this paper, we introduce and study the concept of CS-Rickart modules, that is a module analogue of the concept of ACS rings. A ring R is called a right weakly semihereditary ring if every its finitly generated right ideal is of the form P ⊕ S, where P R is a projective module and S R is a singular module. We describe the ring R over which Mat n (R) is a right ACS ring for any n ∈ N. We show that every finitely generated projective right R-module will to be a CS-Rickart module, is precisely when R is a right weakly semihereditary ring. Also, we prove that if R is a right weakly semihereditary ring, then every finitely generated submodule of a projective right R-module has the form P 1 ⊕ . . . ⊕ P n ⊕ S, where every P 1 , . . . , P n is a projective module which is isomorphic to a submodule of R R , and S R is a singular module. As corollaries we obtain some well-known properties of Rickart modules and semihereditary rings.

Modules close to SSP- and SIP-modules

“…M is called a SSP-lifting module if A i lies above a direct summand of M for all i ∈ I, I is a finite index set, then i∈I A i lies above a direct summand of M (see [23]). The class of CS-Rickart (d-CS-Rickart, SIP-CS, SSP-lifting) modules is studied by the authors in [1,2]. Proof.…”