n-absorbing ideal factorization in commutative rings

Abstract

In this article, we show that Mori domains, pseudo-valuation domains, and $n$-absorbing ideals, the three seemingly unrelated notions in commutative ring theory, are interconnected. In particular, we prove that an integral domain $R$ is a Mori locally pseudo-valuation domain if and only if each proper ideal of $R$ is a finite product of 2-absorbing ideals of $R$. Moreover, every ideal of a Mori locally almost pseudo-valuation domain can be written as a finite product of 3-absorbing ideals. To provide concrete examples of such rings, we study rings of the form $A+XB[X]$ where $A$ is a subring of a commutative ring $B$ and $X$ is indeterminate, which is of independent interest, and along with several characterization theorems, we prove that in such a ring, each proper ideal is a finite product of $n$-absorbing ideals for some $n\ge 2$ if and only if $A$ and $B$ are both Artinian reduced rings and the contraction map $\text{Spec}(B)\to\text{Spec}(A)$ is a bijection. A complete description of when an order of a quadratic number field is a locally pseudo valuation domain, a locally almost pseudo valuation domain or a locally conducive domain is given.

Theorems & Definitions (94)

• Definition 1
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• Theorem 6
• Lemma 7