Some results on 1-absorbing primary and weakly 1-absorbing primary ideals of commutative rings
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Let $R$ be a commutative ring with identity. A proper ideal $I$ of $R$ is called $1$-absorbing primary (\cite{Bad}) if for all nonunit $a,b,c \in R$ such that $abc \in I$, then either $ab \in I$ or $c \in \sqrt{I}$. The concept of $1$-absorbing primary ideals in a polynomial ring, in a PID and in idealization of a module is studied. Moreover, we introduce weakly $1$-absorbing primary ideals which are generalization of weakly prime ideals and $1$-absorbing primary ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing primary if for all nonunit $a,b,c \in R$ such that $0\neq abc \in I$, then either $ab \in I$ or $c \in \sqrt{I}$. Some properties of weakly $1$-absorbing primary ideals are investigated. For instance, weakly $1$-absorbing primary ideals in decomposable rings are characterized. Among other things, it is proved that if $I$ is a weakly $1$-absorbing primary ideal of a ring $R$ and $0 \neq I_1I_2I_3 \subseteq I$ for some ideals $I_1, I_2, I_3$ of $R$ such that $I$ is free triple-zero with respect to $I_1I_2I_3$, then $ I_1I_2 \subseteq I$ or $I_3\subseteq I$.
Abstract
Let $R$ be a commutative ring with identity. A proper ideal $I$ of $R$ is called $1$-absorbing primary (\cite{Bad}) if for all nonunit $a,b,c \in R$ such that $abc \in I$, then either $ab \in I$ or $c \in \sqrt{I}$. The concept of $1$-absorbing primary ideals in a polynomial ring, in a PID and in idealization of a module is studied. Moreover, we introduce weakly $1$-absorbing primary ideals which are generalization of weakly prime ideals and $1$-absorbing primary ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing primary if for all nonunit $a,b,c \in R$ such that $0\neq abc \in I$, then either $ab \in I$ or $c \in \sqrt{I}$. Some properties of weakly $1$-absorbing primary ideals are investigated. For instance, weakly $1$-absorbing primary ideals in decomposable rings are characterized. Among other things, it is proved that if $I$ is a weakly $1$-absorbing primary ideal of a ring $R$ and $0 \neq I_1I_2I_3 \subseteq I$ for some ideals $I_1, I_2, I_3$ of $R$ such that $I$ is free triple-zero with respect to $I_1I_2I_3$, then $ I_1I_2 \subseteq I$ or $I_3\subseteq I$.
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