强乘法集、幂等局部化与$S$-素现象
Strongly multiplicative sets, idempotent localizations, and $S$-prime phenomena
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中文总结 AI 辅助
本文研究交换环的强乘法集，刻画其与幂等局部化及$S$-素理想的关系，并给出保持性结果、挠理论及Krull分离引理等应用。
中文摘要 AI 辅助
交换环$R$的乘法集$S$称为强乘法的，如果$S$中元素的每个族$(s_i)_{i \in I}$在$S \cap \bigcap_{i \in I} s_iR$中都有一个公共倍数。我们将强乘法集的结构结果与其素理论和模理论应用相结合。我们证明$S$是强乘法的当且仅当在$S$处的局部化与理想的任意交交换，当且仅当$R_S$是某个幂等元的局部化，当且仅当$D(S)$在$\Spec(R)$中是既开又闭的。然后我们发展了在同态、乘积、因子环、平凡扩张和融合下的保持性结果；描述了相关的分裂挠理论；证明了几乎乘法集在其乘法包络之外不产生新的情形；并记录了有限生成模中子模交的Mittag--Leffler细化。在素理论方面，我们将强乘法集与强素理想和强零维环联系起来，回答了Hamed--Malek关于强乘法性在$S$-素理想链中作用的问题，证明了强Krull分离引理以及$R_S$的极大理想对应，并将该理论与正则$m$-补算子联系起来。特别地，每个非平凡的强乘法局部化必须反转一个零因子。
英文摘要
A multiplicative set $S$ of a commutative ring $R$ is strongly multiplicative if every family $(s_i)_{i \in I}$ of elements of $S$ admits a common multiple in $S \cap \bigcap_{i \in I} s_iR$. 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.
发表机构
- Hoseo University（湖西大学）
- Marmara University（马尔马拉大学）
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