Additive subgroups of a module that are saturated with respect to a subset of the ring

TL;DR

This paper introduces and studies $T$-factroids, a class of additive subgroups of modules saturated with respect to a subset of a ring, connecting them to various algebraic structures and concepts.

Contribution

It defines $T$-factroids, explores their properties, and relates them to zero-divisors, prime ideals, Euclidean domains, and introduces sublocalizable rings as a generalization.

Findings

$T$-factroids generalize $T$-submodules with dual properties.

Connections established between $T$-factroids and zero-divisors, prime ideals, and Euclidean domains.

Introduction of sublocalizable rings as a unifying concept.

Abstract

Let $T$ be a subset of a ring $A$, and let $M$ be an $A$-module. We study the additive subgroups $F$ of $M$ such that, for all $x \in M$, if $tx \in F$ for some $t \in T$, then $x \in F$. We call any such subset $F$ of $M$ a $T$-factroid of $M$, which is a kind of dual to the notion of a $T$-submodule of $M$. We connect the notion with the zero-divisors on $M$, various classes of primary and prime ideals of $A$, Euclidean domains, and the recent concepts of unit-additive commutative rings and of Egyptian fractions with respect to a multiplicative subset of a commutative ring. We also introduce a common generalization of local rings and unit-additive rings, called *sublocalizable* rings, and relate them to $T$-factroids.