Note on weakly 1-absorbing primary ideals

Abstract

An ideal I of a commutative ring R is called a weakly primary ideal of R if whenever a,b ? R and 0 ? ab ? I, then a ? I or b ? ?I. An ideal I of R is called weakly 1-absorbing primary if whenever nonunit elements a, b, c ? R and 0 ? abc ? I, then ab ? I or c ? ?I. In this paper, we characterize rings over which every ideal is weakly 1-absorbing primary (resp. weakly primary). We also prove that, over a non-local reduced ring, every weakly 1-absorbing primary ideals is weakly primary.