On the irreducible components of an ideal

Abstract

Let I be an M -primary ideal in a local ring (R, M) and let irr(I) denote the set of irreducible components of I, where an ideal q is an irreducible component of I if q occurs as a factor in some decomposition of I as an irredundant intersection of irreducible ideals. We give several characterizations of the ideals in irr(I) and show that if J is an ideal between I and an irreducible component of I, then J is the intersection of ideals in irr(I). We also exhibit examples showing that there may exist irreducible ideals containing I that contain no ideal in irr(I). Also, we determine necessary and sufficient conditions that the pricipal ideal uR[u, tI] of the Rees ring R[u, tI] have a unique cover, and apply this to the study of the form ring of R with respect to I.