Generalizations of 2-absorbing principally right primary ideals in non-commutative rings
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Let $R$ be a non-commutative ring with $1\neq0$ and $\mathit{S}(R)$ be the set of all ideals of $R$. In this paper, we extend the concept of 2-absorbing principally right primary ideals to the context of $\phi$-2-absorbing principally right primary ideals. Let $\phi$ : $\mathit{S}(R)$ $\rightarrow \mathit{S}(R)\cup\{\emptyset\}$ be a function. A proper ideal $I$ of $R$ is said to be a $\phi-2$-absorbing principally right primary ideal of $R$ if whenever $a,b,c\in R$ with $aRbRc\subseteq I$ and $aRbRc\nsubseteq\phi(I)$ implies $ab\in I$ or $ac\in\sqrt{I}$ or $bc\in\sqrt{I.}$ A number of results and characterizations concerning $\phi$-2-absorbing principally right primary ideals are given.
Abstract
Let $R$ be a non-commutative ring with $1\neq0$ and $\mathit{S}(R)$ be the set of all ideals of $R$. In this paper, we extend the concept of 2-absorbing principally right primary ideals to the context of $\phi$-2-absorbing principally right primary ideals. Let $\phi$ : $\mathit{S}(R)$ $\rightarrow \mathit{S}(R)\cup\{\emptyset\}$ be a function. A proper ideal $I$ of $R$ is said to be a $\phi-2$-absorbing principally right primary ideal of $R$ if whenever $a,b,c\in R$ with $aRbRc\subseteq I$ and $aRbRc\nsubseteq\phi(I)$ implies $ab\in I$ or $ac\in\sqrt{I}$ or $bc\in\sqrt{I.}$ A number of results and characterizations concerning $\phi$-2-absorbing principally right primary ideals are given.