The Equality (A∩B)n=An∩Bnfor Ideals

Abstract

LetDbe an integral domain with identity, and letRbe a commutative ring. Ifnis a positive integer,Rwill be said tohave property (n), (n)′,or(n)″according asproperty(n):For anyx, y∊R,(x,y)n= (xn, yn).property(n)′: For anyx ∊ Rand any idealAofRsuch that xn∊ An, it follows thatx ∊ A.property(n)′: For any idealsA, BofR, (A∩ B)n= An∩ Bn.J. Ohm introduced property (n) in [7] in connection with the question: Ifn≦ 2 and ifDhas property (n), mustDbe a Prüfer domain? (The integral domainDwith identity is a Prüfer domain if each nonzero finitely generated ideal ofDis invertible; equivalently,DPis a valuation ring for each proper prime idealPofD.)Prior to Ohm's paper, it was known that ifDhas property (2) and ifDis integrally closed, thenDis Prüfer.