On Pseudo 2-prime Ideals and Almost Valuation Domains
Bull. Korean Math. Soc. Published online May 10, 2021
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∈P for some x,y∈R, then either x²ⁿ∈Pⁿ or y²ⁿ∈Pⁿ for some n∈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