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arXiv · 2501.11575

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

Jesse Elliott·Neil Epstein
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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.

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Jesse Elliott, Neil Epstein. 2025-01-20. Additive subgroups of a module that are saturated with respect to a subset of the ring. https://arxiv.org/abs/2501.11575

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