Narrow Results

A module $M$ is called coseparable ($𝔪$-coseparable) if for every submodule $U$ of $M$ such that $M/U$ is finitely generated ($M/U$ is simple), there exists a direct summand $V$ of $M$ such that $V\subseteq U$ and $M/V$ is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ($𝔪$-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of $𝔪$-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ($𝔪$-coseparable).

A module $M$ is called ${𝒜}$-separable if every proper finitely generated submodule of $M$ is contained in a proper finitely generated direct summand of $M$. Indecomposable ${𝒜}$-separable modules are shown to be exactly the simple modules. While direct summands of an ${𝒜}$-separable module do not inherit the property, in general, the question of the stability under direct sums is unanswered. But we obtain some partial answers. It is shown that any infinite direct sum of ${𝒜}$-separable modules is ${𝒜}$-separable. Also, we prove that if $M_1$ and $M_2$ are ${𝒜}$-separable modules such that $M_1$ is $M_2$-projective, then $M_1\oplus M_2$ is ${𝒜}$-separable. We conclude the paper by providing some characterizations of several classes of rings in terms of ${𝒜}$-separable modules. Among others, we prove that the class of rings $R$ for which every (injective) $R$-module is ${𝒜}$-separable is exactly that of semisimple rings.