Abstract

“Unique factorization” was central to the initial development of ideal theory. We update this topic with several new results concerning notions of “unique ideal factorization rings” with zero divisors. Along the way, we obtain new characterizations of several well-known kinds of rings in terms of their ideal factorization properties and examine when monoid rings satisfy various kinds of “unique ideal factorization.” Our results include necessary and sufficient conditions for a monoid ring $R[S]$ with S cancellative to be a π-ring, a higher-dimensional generalization of Hardy and Shores’s classic characterization of when $R[S]$ is a general Zerlegung Primideale ring.

摘要

“唯一因式分解”是理想理论最初发展的核心。我们用关于零因数的“唯一理想因式分解环”的概念的几个新结果更新了此主题。在此过程中，我们从几种理想的环的理想因式分解特性方面获得了新的表征，并研究了单曲面环何时满足各种“唯一理想因式分解”。我们的结果包括一个半同义环的必要和充分条件$\mathrm{[R}[\mathrm{小号}]$其中S可以是π环，是Hardy和Shores对when的经典刻画的高维概括$\mathrm{[R}[\mathrm{小号}]$ 是普通的Zerlegung Primideale戒指。