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 ≥ 2 if and only if A ⊆ B is essentially a finite product of field extensions. 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.