On $\phi$-$n$-absorbing primary ideals of commutative rings
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All rings are commutative with $1\neq0$ and $n$ is a positive integer. Let $\phi:\mathfrak{J}(R)\rightarrow \mathfrak{J}(R)\cup\{\emptyset\}$ be a function where $\mathfrak{J}(R)$ denotes the set of all ideals of $R$. We say that a proper ideal $I$ of $R$ is {\it $\phi$-$n$-absorbing primary} if whenever $a_1,a_2,\dots,a_{n+1}\in R$ and $a_1a_2\cdots a_{n+1}\in I\backslash\phi(I)$, either $a_1a_2\cdots a_n\in I$ or the product of $a_{n+1}$ with $(n-1)$ of $a_1,\dots,a_n$ is in $\sqrt{I}$. The aim of this paper is to investigate the concept of $\phi$-$n$-absorbing primary ideals.
Abstract
All rings are commutative with $1\neq0$ and $n$ is a positive integer. Let $\phi:\mathfrak{J}(R)\rightarrow \mathfrak{J}(R)\cup\{\emptyset\}$ be a function where $\mathfrak{J}(R)$ denotes the set of all ideals of $R$. We say that a proper ideal $I$ of $R$ is {\it $\phi$-$n$-absorbing primary} if whenever $a_1,a_2,\dots,a_{n+1}\in R$ and $a_1a_2\cdots a_{n+1}\in I\backslash\phi(I)$, either $a_1a_2\cdots a_n\in I$ or the product of $a_{n+1}$ with $(n-1)$ of $a_1,\dots,a_n$ is in $\sqrt{I}$. The aim of this paper is to investigate the concept of $\phi$-$n$-absorbing primary ideals.
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