ON DEPTH ONE PRIME DIVISORS OF ZERO

Let (R, M) be a local (Noetherian) ring with completion R such that t he maximal ideal M is regular (i.e. Al contains a non-zerodivisor.) Th is 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(MR*/q) = 1.) if and only if M is an associated prim e divisor of every regular ideal I of R contained in a proper principa l ideal.