Strongly multiplicative sets, idempotent localizations, and S-prime phenomena
Abstract
A multiplicative set S of a commutative ring R is strongly multiplicative if every family (si)i ∈ I of elements of S admits a common multiple in S ∩ ⋂i ∈ I siR. We combine the structural results on strongly multiplicative sets with their prime-theoretic and module-theoretic applications. We prove that S is strongly multiplicative if and only if localization at S commutes with arbitrary intersections of ideals, if and only if RS is the localization at an idempotent, and if and only if D(S) is clopen in Spec(R). We then develop permanence results under homomorphisms, products, factor rings, trivial extensions, and amalgamations; describe the associated split torsion theory; show that almost multiplicative sets contribute no new cases beyond their multiplicative hull; and record a Mittag--Leffler refinement for intersections of submodules in finitely generated modules. On the prime-theoretic side, we relate strongly multiplicative sets to strongly prime ideals and strongly zero-dimensional rings, answer the Hamed--Malek question on the role of strong multiplicativity for chains of S-prime ideals, prove a strong Krull separation lemma together with a maximal-ideal correspondence for RS, and connect the theory with the regular m-complement operator. In particular, every nontrivial strongly multiplicative localization must invert a zero divisor.
Related
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BibTeX
@misc{kim2026strongly,
title = {{Strongly multiplicative sets, idempotent localizations, and $S$-prime phenomena}},
author = {Hwankoo Kim and Suat Koç},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16741},
eprint = {2609.16741},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16741},
}
RIS
TY - UNPB
TI - Strongly multiplicative sets, idempotent localizations, and S-prime phenomena
AU - Hwankoo Kim
AU - Suat Koç
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16741
UR - https://arxiv.org/abs/2609.16741
AB - A multiplicative set S of a commutative ring R is strongly multiplicative if every family (s_i)_i ∈ I of elements of S admits a common multiple in S ∩ ⋂_i ∈ I s_iR. We combine the structural results on strongly multiplicative sets with their prime-theoretic and module-theoretic applications. We prove that S is strongly multiplicative if and only if localization at S commutes with arbitrary intersections of ideals, if and only if R_S is the localization at an idempotent, and if and only if D(S) is clopen in Spec(R). We then develop permanence results under homomorphisms, products, factor rings, trivial extensions, and amalgamations; describe the associated split torsion theory; show that almost multiplicative sets contribute no new cases beyond their multiplicative hull; and record a Mittag--Leffler refinement for intersections of submodules in finitely generated modules. On the prime-theoretic side, we relate strongly multiplicative sets to strongly prime ideals and strongly zero-dimensional rings, answer the Hamed--Malek question on the role of strong multiplicativity for chains of S-prime ideals, prove a strong Krull separation lemma together with a maximal-ideal correspondence for R_S, and connect the theory with the regular m-complement operator. In particular, every nontrivial strongly multiplicative localization must invert a zero divisor.
ER -