On strongly multiplicative sets
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918392949112832 |
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| author | Koç, Suat |
| author_facet | Koç, Suat |
| contents | A multiplicative subset $S$ of a ring $R$ is called \textit{strongly multiplicative} if $(\bigcap_{i\inΔ}s_iR)\cap S \neq \emptyset$ for each family $(s_i)_{i\inΔ}$ of elements in $S$. In this paper, we investigate how these sets help stabilize localization and ideal operations. We show that localization and arbitrary intersections commute, meaning $S^{-1}(\bigcap I_α) = \bigcap S^{-1}I_α$ for any family of ideals, if and only if $S$ is strongly multiplicative. Furthermore, we characterize some important classes of rings, such as total quotient rings and strongly zero-dimensional rings, in terms of strongly multiplicative sets. We also answer an open question by Hamed and Malek about whether this condition is needed for the existence of $S$-minimal primes. Furthermore, we demonstrate that if $S$ is a strongly multiplicative set and $S \not\subseteq U(R)$, then $S$-minimal primes are not classical prime ideals, and we provide an algorithmic approach to constructing such ideals. Finally, we prove a Strong Krull's Separation Lemma, which guarantees a maximal ideal disjoint from $S$. As an application of Strong Krull's Separation Lemma, we establish a one-to-one correspondence between the maximal ideals of $S^{-1}R$ and the maximal ideals of $R$ disjoint from a strongly multiplicative set $S$ of $R$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23935 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On strongly multiplicative sets Koç, Suat Commutative Algebra Rings and Algebras 13A15, 13A99, 13B30, 13G05 A multiplicative subset $S$ of a ring $R$ is called \textit{strongly multiplicative} if $(\bigcap_{i\inΔ}s_iR)\cap S \neq \emptyset$ for each family $(s_i)_{i\inΔ}$ of elements in $S$. In this paper, we investigate how these sets help stabilize localization and ideal operations. We show that localization and arbitrary intersections commute, meaning $S^{-1}(\bigcap I_α) = \bigcap S^{-1}I_α$ for any family of ideals, if and only if $S$ is strongly multiplicative. Furthermore, we characterize some important classes of rings, such as total quotient rings and strongly zero-dimensional rings, in terms of strongly multiplicative sets. We also answer an open question by Hamed and Malek about whether this condition is needed for the existence of $S$-minimal primes. Furthermore, we demonstrate that if $S$ is a strongly multiplicative set and $S \not\subseteq U(R)$, then $S$-minimal primes are not classical prime ideals, and we provide an algorithmic approach to constructing such ideals. Finally, we prove a Strong Krull's Separation Lemma, which guarantees a maximal ideal disjoint from $S$. As an application of Strong Krull's Separation Lemma, we establish a one-to-one correspondence between the maximal ideals of $S^{-1}R$ and the maximal ideals of $R$ disjoint from a strongly multiplicative set $S$ of $R$. |
| title | On strongly multiplicative sets |
| topic | Commutative Algebra Rings and Algebras 13A15, 13A99, 13B30, 13G05 |
| url | https://arxiv.org/abs/2512.23935 |