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 $P\oplus S,$ where $P_R$ is a projective module and $S_R$ is a singular module. We describe the ring $R$ over which ${\mathrm{Mat}}_n(R)$ is a right ACS ring for any $n\in \mathbb{N}$. 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 $P_1\oplus \dots \oplus P_n\oplus S$, where every $P_1,\dots ,P_n$ 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.

17 pages