On pseudo 2-prime ideals and almost valuation domains
Bull. Korean Math. Soc. 2021 Vol. 58, No. 4, 897-908
Published online May 10, 2021
Printed July 31, 2021
Suat Ko\c{c}
Marmara University
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).
Keywords : Prime ideal, 2-prime ideal, pseudo 2-prime ideal, valuation domain, almost valuation domain
MSC numbers : Primary 13A15, 13F30, 13G05
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