Additive subgroups of a module that are saturated with respect to a subset of the ring
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916874292297728 |
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| author | Elliott, Jesse Epstein, Neil |
| author_facet | Elliott, Jesse Epstein, Neil |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11575 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Additive subgroups of a module that are saturated with respect to a subset of the ring Elliott, Jesse Epstein, Neil Rings and Algebras Commutative Algebra 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. |
| title | Additive subgroups of a module that are saturated with respect to a subset of the ring |
| topic | Rings and Algebras Commutative Algebra |
| url | https://arxiv.org/abs/2501.11575 |