On pseudo 2-prime ideals and almost valuation domains
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In this paper, we introduce the notion of pseudo 2-prime ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity. A proper ideal $P$ of $R$ is said to be a pseudo 2-prime ideal if whenever $xy\in P$ for some $x,y\in R$, then $x^{2n}\in P^{n}$ or $y^{2n}\in P^{n}$ for some $n\in \mathbb{N}$. Various examples and properties of pseudo 2-prime ideals are given. We also characterize pseudo 2-prime ideals of PID's and von Neumann regular rings. Finally, we use pseudo 2-prime ideals to characterize almost valuation domains (AV-domains).
Abstract
In this paper, we introduce the notion of pseudo 2-prime ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity. A proper ideal $P$ of $R$ is said to be a pseudo 2-prime ideal if whenever $xy\in P$ for some $x,y\in R$, then $x^{2n}\in P^{n}$ or $y^{2n}\in P^{n}$ for some $n\in \mathbb{N}$. Various examples and properties of pseudo 2-prime ideals are given. We also characterize pseudo 2-prime ideals of PID's and von Neumann regular rings. Finally, we use pseudo 2-prime ideals to characterize almost valuation domains (AV-domains).
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