An idealIin a commutative ringRis called az°-ideal ifIconsists of zero-divisors and for eacha∈Ithe intersection of all minimal prime ideals containingais contained inI.We prove that in a large class of rings, containing Noetherian reduced rings, Zero-dimensional rings, polynomials over reduced rings andC(X), every ideal consisting of zero-divisors is contained in a primez°-ideal. It is also shown that the classical ring of quotients of a reduced ring is regular if and only if every primez°-ideal is a minimal prime ideal and the annihilator of a f.g. ideal consisting of zero-divisors is nonzero. We observe thatz°-ideals behave nicely under contractions and extensions.