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].