Commutative Rings with n -1-Absorbing Prime Factorization

Let Formula: see text be a commutative ring with Formula: see text and Formula: see text be a fixed positive integer. A proper ideal Formula: see text of Formula: see text is said to be an Formula: see text-OA ideal if whenever Formula: see text for some nonunits Formula: see text, then Formula: see text or Formula: see text. A commutative ring Formula: see text is said to be an Formula: see text-OAF ring if every proper ideal Formula: see text of Formula: see text is a product of finitely many Formula: see text-OA ideals. In fact, Formula: see text-OAF rings and Formula: see text-OAF are exactly the general ZPI rings and OAF rings, respectively. In addition to giving various properties of Formula: see text-OAF rings, we give a characterization of Noetherian von Neumann regular rings in terms of our new concept. Furthermore, we investigate the Formula: see text-OAF property of some extension of rings such as the polynomial ring Formula: see text, the formal power series ring Formula: see text, the ring of Formula: see text, and the trivial extension Formula: see text of an Formula: see text-module Formula: see text.