### 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.

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.

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