When mutually subisomorphic Baer modules are isomorphic

TL;DR

This paper investigates when subisomorphic Baer modules are necessarily isomorphic, extending the Schröder-Bernstein property from sets to modules over rings, with a focus on Baer rings and modules.

Contribution

It characterizes conditions under which subisomorphic Baer modules are isomorphic and explores the SB property in extending modules over commutative domains.

Findings

Baer rings satisfy the SB property under certain conditions

Characterization of commutative domains where subisomorphic extending modules are isomorphic

Extension of Schröder-Bernstein property to module theory

Abstract

The Schr\"{o}der-Bernstein Theorem for sets is well known. The question of whether two subisomorphic algebraic structures are isomorphic to each other, is of interest. An $R$-module $M$ is said to satisfy the Schr\"{o}der-Bernstein (or SB) property if any pair of direct summands of $M$ are isomorphic provided that each one is isomorphic to a direct summand of the other. A ring $R$ (with an involution $\star$) is called a Baer (Baer $\star$-)ring if the right annihilator of every nonempty subset of $R$ is generated by an idempotent (a projection). It is clear that every Baer $\star$-ring is a Baer ring. Kaplansky showed that Baer $\star$-rings satisfy the SB property. This motivated us to investigate whether any Baer ring satisfies the SB property. In this paper we carry out a study of this question and investigate when two subisomorphic Baer modules are isomorphic. Besides, we study extending modules which satisfy the SB property. We characterize a commutative domain $R$ over which any pair of subisomorphic extending modules are isomorphic.