CS-Rickart modules

TL;DR

This paper introduces CS-Rickart modules, explores their relation to right weakly semihereditary rings, and characterizes modules and rings with these properties, extending known results in module and ring theory.

Contribution

It defines CS-Rickart modules, characterizes rings where finitely generated projective modules are CS-Rickart, and links these concepts to right weakly semihereditary rings.

Findings

Finitely generated projective modules are CS-Rickart iff the ring is right weakly semihereditary.

Every finitely generated submodule of a projective module decomposes into projective and singular parts.

Characterizations of rings where matrix rings are right ACS rings for all sizes.

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 $P_{1}\oplus \ldots\oplus P_{n}\oplus S$, where every $P_{1}, \ldots, P_{n}$ 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.