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Keywords:
strongly 1-absorbing primary ideal; $n$-ideal; primary ideal; semi-primary ideal
strongly 1-absorbing primary ideal; $n$-ideal; primary ideal; semi-primary ideal
Summary:
Let $R$ be a commutative ring with identity. We study the concept of strongly \hbox {1-absorbing} primary ideals which is a generalization of $n$-ideals and a subclass of $1$-absorbing primary ideals. A proper ideal $I$ of $R$ is called strongly 1-absorbing primary if for all nonunit elements $a,b,c \in R$ such that $abc \in I$, it is either $ab \in I$ or $c \in \sqrt {0}$. Some properties of strongly 1-absorbing primary ideals are studied. Finally, rings $R$ over which every semi-primary ideal is strongly 1-absorbing primary, and rings $R$ over which every strongly \hbox {1-absorbing} primary ideal is prime (or primary) are characterized. Many examples are given to illustrate the obtained results.
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