s.Baer and s.Rickart Modules

TL;DR

This paper explores the properties of s.Baer modules, establishing their closure properties, connections to projective modules, and implications for ring theory, including torsion theories and projectivity in triangular matrix rings.

Contribution

It introduces and analyzes s.Baer modules, linking them to projective modules and ring properties, with new results on their structure and applications in ring theory.

Findings

s.Baer modules satisfy various closure properties

Every s.Baer module is an essential extension of a projective module

In triangular matrix rings, s.Baer submodules are projective

Abstract

In this paper, we study module theoretic definitions of the Baer and related ring concepts. We say a module is s.Baer if the right annihilator of a nonempty subset of the module is generated by an idempotent in the ring. We show that s.Baer modules satisfy a number of closure properties. Under certain conditions, a torsion theory is established for the s.Baer modules, and we provide examples of s.Baer torsion modules and modules with a nonzero s.Baer radical. The other principal interest of this paper is to provide explicit connections between s.Baer modules and projective modules. Among other results, we show that every s.Baer module is an essential extension of a projective module. Additionally, we prove, with limited and natural assumptions, that in a generalized triangular matrix ring every s.Baer submodule of the ring is projective. As an application, we show that every prime ring with a minimal right ideal has the strong summand intersection property. Numerous examples are provided to illustrate, motivate, and delimit the theory.