Commutative rings with one-absorbing factorization

Abstract

Let R be a commutative ring with nonzero identity. Yassine et al. defined the concept of 1-absorbing prime ideals as follows: a proper ideal I of R is said to be a 1-absorbing prime ideal if whenever xyz 2 I for some nonunit elements x, y, z 2 R, then either xy 2 I or z 2 I: We use the concept of 1absorbing prime ideals to study those commutative rings in which every proper ideal is a product of 1-absorbing prime ideals (we call them OAFrings). Any OAF-ring has dimension at most one and local OAF-domains (D, M) are atomic such that M 2 is universal.

Classical 1-Absorbing Primary Submodules

“…Thus, ℜ admits a 1-a.p ideal that is not primary. Then by [22] (Lemma 2.1), ℜ is a local ring with unique maximal ideal q of ℜ, such that q 2 ⊆ (K : ℜ η) for some η ∈ M − K.…”