Good ideals in Gorenstein local rings

Let I be an m-primary ideal in a Gorenstein local ring (A,m) with dim A = d , and assume that I contains a parameter ideal Q in A as a reduction. We sa y that I is a good ideal in A if G = Sigma (n greater than or equal to0) I- n / In+1 is a Gorenstein ring with a(G) = 1 d. The associated graded ring G of I is a Gorenstein ring with a(G) = d if and only if I = Q. Hence good i deals in our sense are good ones next to the parameter ideals Q in A. A bas ic theory of good ideals is developed in this paper. We have that I is a go od ideal in A if and only if I-2 = QI and I = Q : I. First a criterion for finite-dimensional Gorenstein graded algebras A over fields k to have nonem pty sets X-A of good ideals will be given. Second in the case where d = 1 w e will give a correspondence theorem between the set X-A and the set Y-A of certain overrings of A. A characterization of good ideals in the case wher e d = 2 will be given in terms of the goodness in their powers. Thanks to K ato's Riemann-Roch theorem, we are able to classify the good ideals in two- dimensional Gorenstein rational local rings. As a conclusion we will show t hat the structure of the set X-A of good ideals in A heavily depends on d = dim A. The set X-A may be empty if d less than or equal to 2, while X-A is necessarily infinite if d greater than or equal to 3 and A contains a fiel d. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring k[X-1,X-2,X-3] in three variables over a field k. Examples are given to illustrate the theorems.