发表机构
Stellenbosch University; University of Minho（斯泰伦博斯大学; 米尼奥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带零幺半群直积的因子重构，提出基于可补中心幂等元偏序的乘法重构机制，证明同构为单项式并给出自同构群的圈积描述，应用于不可分解酉环及剩余类环。
AI 中文摘要
我们研究带零幺半群的直积，并给出一个准则，据此可仅从乘法结构恢复其因子。经典分解理论通过因子同余、中心元素和细化性质来刻画直积，而我们给出一个具体的乘法重构机制。若因子没有非平凡的可补中心幂等元，则坐标幂等元恰为可补中心幂等元偏序集的原子和余原子，且其乘法稳定化子恰为坐标因子和余因子。由此可知，此类直积之间的每个同构都是单项式的，从而得到自同构群的相应圈积描述。更一般地，每个积分解都由对原始因子分组得到；特别地，严格细化成立。我们将这些结果应用于直接不可分解酉环（包括连通交换环）的乘法幺半群，以及由剩余类环产生的算术例子。
英文摘要
We study direct products of monoids with zero and give a criterion under which their factors can be recovered from the multiplicative structure alone. While classical decomposition theory encodes direct products through factor congruences, central elements, and refinement properties, we give a concrete multiplicative reconstruction mechanism. If the factors have no nontrivial complemented central idempotents, then the coordinate idempotents are precisely the atoms and coatoms of the complemented-central-idempotent poset, and their multiplicative stabilizers are precisely the coordinate factors and cofactors. It follows that every isomorphism between such products is monomial, yielding the corresponding wreath-product description of automorphism groups. More generally, every product decomposition is obtained by grouping the original factors; in particular, strict refinement follows. We apply these results to multiplicative monoids of directly indecomposable unital rings, including connected commutative rings, and to arithmetic examples arising from residue-class rings.
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