On ideals consisting entirely of zero divisors

Abstract

An ideal Iin a commutative ring Ris called a z°-ideal if Iconsists of zero-divisors and for each a∊ Ithe intersection of all minimal prime ideals containing ais contained in I.We prove that in a large class of rings, containing Noetherian reduced rings, Zero-dimensional rings, polynomials over reduced rings and C(X), every ideal consisting of zero-divisors is contained in a prime z°-ideal. It is also shown that the classical ring of quotients of a reduced ring is regular if and only if every prime z°-ideal is a minimal prime ideal and the annihilator of a f.g. ideal consisting of zero-divisors is nonzero. We observe that z°-ideals behave nicely under contractions and extensions.

^∗The first two authors are partially supported by Institute for Studies in Theoretical Physics and Mathematics (IPM)

^∗The first two authors are partially supported by Institute for Studies in Theoretical Physics and Mathematics (IPM)

Notes

^∗The first two authors are partially supported by Institute for Studies in Theoretical Physics and Mathematics (IPM)