Simple-separable modules

Year 2024, , 29 - 50, 12.07.2024

Rachid Ech-chaouy Rachid Trıbak

https://doi.org/10.24330/ieja.1449459

Abstract

A module $M$ over a ring is called simple-separable if every simple submodule of $M$ is contained in a finitely generated
direct summand of $M$. While a direct sum of any family of simple-separable modules is shown to be always simple-separable, we prove that
a direct summand of a simple-separable module does not inherit the property, in general. It is also shown that an injective module $M$
over a right noetherian ring is simple-separable if and only if $M=M_1 \oplus M_2$ such that $M_1$ is separable and $M_2$ has zero socle.
The structure of simple-separable abelian groups is completely described.

Keywords

Separable module, simple-separable module, V-ring, $\pi$-V-ring

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