Suppose that R is a power series ring over a field, or a ring of convergent power series, with maximal ideal m. Given two ideals I, J in R, it is of interest to determine general conditions ensuring that there is an isomorphism RrI ( RrJ.
w x ลฝ In CS see Theorem 1.6 of this paper for the extension to the analytic . case , we proved that if I is reduced and equidimensional, and H is an ideal defining the nonsmooth locus of RrI, then there exists an integer n such that if J is equidimensional of the same height as I, and I ' J mod ลฝ . n j I , then there exists an isomorphism RrI ( RrJ. This extends a theow x rem on isolated singularities of Hironaka H1 .
In this paper we obtain generalizations to arbitrary ideals.
The ideal theorem would be that if H is an ideal defining the nonsmooth locus of RrI, then there exists an integer n such that if J is any ideal with J ' I mod H n , then RrI and RrJ are isomorphic. However, it is easy to see that this cannot be true, since this equivalence condition can be fulfilled, while the height of J is larger than that of I.
ลฝ n . Suppose that F g Hom R , R is a homomorphism and I is the image of F. Let H be a proper ideal of R. F is said to be finitely determined by H if there is a positive integer a, such that F ( G mod H a implies that there exist an automorphism of R and an R-isomorphism A of R n such ลฝ . that F s AG. Thus, if F is finitely determined by an ideal H, which depends on I, then we get an intrinsic criterion for determining when RrI would be isomorphic to another singularity RrJ. The idea of finite deterw x mination was introduced by Mather in M where he proved necessary and sufficient conditions for existence of a nontrivial finite determination for analytic and C ฯฑ mappings. In this paper, we extend these ideas and results. *Both authors are partially supported by NSF. 16