Results on n-Absorbing Ideals of Commutative Rings

RESULTS ON N-ABSORBING IDEALS OF COMMUTATIVE RINGS by Alison Elaine Becker The University of Wisconsin-Milwaukee, 2015 Under the Supervision of Dr. Allen Bell Let R be a commutative ring with 1 6= 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 [1] to a concept called n-absorbing ideals. A proper ideal I of R is said to be an n-absorbing ideal if whenever x1 · · ·xn+1 ∈ I for x1, · · · , xn+1 ∈ R then there are n of the xi’s whose product is in I. This paper will provide proofs of several properties in [1] which are stated without proof, and will study how several theorems from 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 [3], 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 x1, . . . , xn+1 ∈ R and x1x2 · · ·xn+1 ∈ I then either x1x2 · · ·xn ∈ I or a product of n of the xis (other than x1 · · ·xn) 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. We will also show that for R = Z and I = nZ if n has k prime factors, then I is a k-absorbing primary ideal, and is not, in fact, a (k − 1)-absorbing primary ideal of R. This will lead us to a conclusion about the intersection of ideals of this form.