Results on n-Absorbing Ideals of Commutative Rings

Abstract

Let R be a commutative ring with n≥0. In his paper On 2-absorbing Ideals of Commutative Rings, Ayman Badawi introduces a generalization of prime ideals called 2-absorbing ideals, and this idea is further generalized in a paper by Anderson and Badawi to a concept called n-absorbing ideals. A proper ideal I of R is said to be an n-absorbing ideal if whenever x_1…x_(n+1) ∈I for x_1,…,x_(n+1 )∈R then there are n of the x_i's whose product is in I. This paper will provide proofs of several properties in Badawi’s paper which are stated without proof, and will study how several theorems in Badawi’s initial paper on 2-absorbing ideals can be extended to n-absorbing ideals of R. Additionally, Badawi introduces a generalization of primary ideals in his paper On 2-absorbing Primary Ideals in Commutative Rings, and this paper generalizes that idea further by defining n-absorbing primary ideals of R. Let n be a positive integer. A proper ideal I of a commutative ring R is said to be an n-absorbing primary ideal of R if whenever x_1,…〖,x〗_(n+1)∈R and x_1…x_(n+1)∈I then either and x_1…x_n∈I or a product of n of the x_i^' s (other than x_1,…〖,x〗_n) is in √I . We will prove several basic properties of n-absorbing primary ideals, including that any n-absorbing primary ideal is m-absorbing for m > n.