On depth one prime divisors of zero

Abstract

Let (R, M) be a local (Noetherian) ring with completion R* such that the maximal ideal Mis regular (i.e. M contains a non-zerodivisor.) This note shows that R* contains a depth one prime divisor of zero (i.e. there exists a prime ideal q in R* such that q is an associated prime of R* with height(M R^*/q) = 1.) if and only if M is an associated prime divisor of every regular ideal I of R contained in a proper principal ideal.