Let X be an infinite cyclic group. An example of two noncommutative nonisomorphic rings R, S such that their group rings RX, SX are isomorphic has been given in [1]. In the present note, we show that there also exist commutative nonisomorphic noetherian domains A, B of Krull dimension 2 such that the group rings AX, BX are isomorphic. That solves Problem 27 of [4] in the negtive. In what follows, some elementary properties of Dedekind domains contained in quadratic extensions of the field of rationals Q will be used (cf. [2]). Let 1 + iV4i a= l 2 ) D= Z[a], K= Q(a). If 4p is the nontrivial automorphism of K then q( is the only nontrivial automorphism of Dedekind domain D and it induces an automorphism of the group of fractional ideals of K. Let I be the ideal in D generated by 2 and a. Then, it may be easily checked that I is not a principal ideal, I n p(I) = 2D and I5 = bD, where b = 4 + a. Thus, if _ means coincidence of classes of fractional ideals, then (I) I4_I-' and p(I-1) I. Hence, I2 is not equivalent to pm(I') where m = 1, 2; 1 = + 1. Now, let m0 00 A= ff I2ntn B = e Intn c K n m -on--o where is an infinite cyclic group which is generated by 1. We shall show that the rings A, B are not isomorphic. Let us suppose a: A -k B is an isomorphism. Under localization at S = Z {0), a becomes an isomorphism of K . Hence a(K) = K and so GIK = rpm where m = 1 or m = 2 (cf. [4]). Moreover, a(t) = cit, c E K*, I = + 1. Since It' = a(I2t) = a(I2)ctl we have a(I2) _I. Thus 4pm(I2) I=, which is impossible. Now, we shall establish an isomorphism of the group rings AX, BX. Let T be the K-automorphism of KKt> given by formulas r(t) = 12x, T(x) = b15x2 where x generates the group X. Then Tr' is given by T-K(t) = b-'C-2x, T-'(x) = b2t5x-2. Since b generates I5 then it may be computed that T(AX) c BX and T-'(BX) c AX. Thus, T induces an isomorphism of the rings AX, BX. Similar examples may be constructed starting from some other Dedekind domains. It is easily seen that the above-considered rings A, B are finitely generated and hence noetherian. Moreover, the Krull dimension of A, B equal to 2. It is known Received by the editors August 12, 1980. 1980 Mathematics Subject Classification Primary 16A27. ? 1981 American Mathematical Society 0002-9939/8 1/0000-0504/$01.50 459 This content downloaded from 207.46.13.60 on Thu, 21 Apr 2016 07:33:28 UTC All use subject to http://about.jstor.org/terms