Collections of an ideal: any n-absorbing ideal is strongly n-absorbing

Abstract

We show that any n-absorbing ideal must be strongly n-absorbing, which is the first of Anderson and Badawi’s three interconnected conjectures on absorbing ideals. We prove this by introducing and studying objects called maximal and semimaximal collections, which are tools for analyzing the multiplicative ideal structure of commutative rings. Furthermore, we discuss and make heavy use of previous work on absorbing ideals done by Anderson and Badawi, Choi and Walker, Laradji, and Donadze, among others. At the end of this paper, we pose three conjectures, each of which implies the last unsolved conjecture made by Anderson and Badawi.