Several different versions of “factoriality” have been defined for commutative rings with zero divisors. We apply semigroup theory to study these notions in the context of a commutative monoid ring R[S], determining necessary and sufficient conditions for R[S] to be various kinds of “unique factorization rings.” Our work generalizes Anderson et al.’s results about “unique factorization” in R[X], Gilmer and Parker’s characterization of factorial monoid domains, and Hardy and Shores’s classification of when R[S] is a principal ideal ring (for S cancellative). Along the way, we determine when R[S] is “restricted cancellative” or satisfies various “(restricted) ideal cancellation laws.”

对于零除数的交换环，已经定义了几种不同的“阶乘”形式。我们应用半群论在可交换的id形环R [ S ]的上下文中研究这些概念，确定R [ S ]成为各种“唯一分解环”的必要条件和充分条件。我们的工作概括了Anderson等人关于R [ X ]中的“唯一因式分解” ，Gilmer和Parker对阶乘半体域的刻画的结果，以及Hardy和Shores对R [ S ]是主要理想环的分类（对于S消去性）的结果。 ）。一路上，我们确定何时R[ S ]是“受限取消”或满足各种“（受限）理想取消法则”。