Bull. Korean Math. Soc. 2021; 58(5): 1069-1078
Online first article August 25, 2021 Printed September 30, 2021
https://doi.org/10.4134/BKMS.b190868
Copyright © The Korean Mathematical Society.
Reza Nikandish, Mohammad Javad Nikmehr, Ali Yassine
Jundi-Shapur University of Technology; K.N. Toosi University of Technology; K.N. Toosi University of Technology
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 $I_3\subseteq I$.
Keywords: 1-absorbing primary ideal, weakly 1-absorbing primary ideal, prime ideal, weakly prime ideal
MSC numbers: Primary 13A15, 13C05
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