Rings over which every semi-primary ideal is 1-absorbing primary

Abstract

Let R be commutative ring with 1≠0. A proper ideal I of R is called a 1-absorbing primary ideal of R if whenever nonunit elements a,b,c∈R and abc∈I, then ab∈I or c∈I. It is proved that every primary ideal of R is 1-absorbing primary and every 1-absorbing primary ideal of R is semi-primary (that is ideals with prime radical). However, these three concepts are different. In this paper, we characterize rings R over which every semi-primary ideal is 1-absorbing primary and (resp. Noetherian) rings R over which every 1-absorbing primary ideal is prime (resp. primary). Many examples are given to illustrate the obtained results.