Quasi J-submodules

TL;DR

This paper extends the concept of quasi J-ideals from rings to modules, providing new characterizations, properties, and classes of modules, especially in finitely generated faithful multiplication modules.

Contribution

It introduces the notion of quasi J-submodules, generalizes presimplifiable modules, and characterizes quasi J-ideals in idealization rings.

Findings

Characterization of quasi J-submodules in finitely generated faithful multiplication modules

Introduction of new classes of modules generalizing presimplifiable modules

Description of quasi J-ideals in idealization rings

Abstract

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. The aim of this paper is to extend the notion of quasi $J$-ideals of commutative rings to quasi $J$-submodules of modules. We call a proper submodule $N$ of $M$ a quasi $J$-submodule if whenever $r\in R$ and $m\in M$ such that $rm\in N$ and $r\notin(J(R)M:M)$, then $m\in M$-$rad(N)$. We present various properties and characterizations of this concept (especially in finitely generated faithful multiplication modules). Furthermore, we provide new classes of modules generalizing presimplifiable modules and justify their relation with (quasi) $J$-submodules. Finally, for a submodule $N$ of $M$ and an ideal $I$ of $R$, we characterize the quasi $J$-ideals of the idealization ring $R(+)M$.