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 $a, b \in  R$ and $0 \neq ab \in  I$, then $a \in  I$
or $b \in \sqrt{I}$. An  ideal $I$ of $R$ is called  weakly 1-absorbing primary    if whenever nonunit elements  $a,b,c\in R$ and $0\neq abc \in I$, then $ab \in I$ or $c \in \sqrt{I}$. 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.