ON PSEUDO 2-PRIME IDEALS AND ALMOST VALUATION DOMAINS - (:unkn) unknown (KOREAN MATHEMATICAL SOC, 2021)
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 is an element of P for some x, y is an element of R, then x(2n) is an element of P-n or y(2n) is an element of P-n for some n is an element of 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; DIVIDED RINGS