A new approach to Baer and dual Baer modules

Abstract

Let $R$ be a ring. It is proved that an $R$-module $M$ is Baer (resp. dual Baer) if and only if every exact sequence $0\to X\to M\to Y\to 0$ with $Y\in$ Cog$(M_R)$ (resp. $X\in$ Gen$(M_R)$) splits. This shows that being (dual) Baer is a Morita invariant property. As more applications, the Baer condition for the $R$-module $M^{+}$ = Hom${}_{\mathbb{Z}}(M,\mathbb{Q}/\mathbb{Z})$ is investigated and shown that $R$ is a von Neumann regular ring, if $R^{+}$ is a Baer $R$-module. Baer modules with (weak) chain conditions are studied and determined when a Baer (resp. dual baer) module is a direct sum of mutually orthogonal prime (resp. co-prime) modules. Finitely generated dual Baer modules over commutative rings are studeid