Let k be an algebraically closed field, and let R--k [[Z, Y]]/(F), where F 6 k [[X, Y]] is an irreducible power series. Gorenstein [6], Th. 6, was apparently the first to remark that gR (S/f)= 2gR (R/[) where S= k [[t]] is the integral closure of R and f=(R: S) s the conductor. Kunz [11] generalized this result to one-dimensional local analytically irreducible rings. On the other hand, Delgado [5] considered algebroid curves having several branches and-under a suitable definition of symmetry of the semigroup of values which is now a subsemigroup of IN h, h being the number of branches-proved a similar result. In this note the result of Delgado will be shown to hold for one-dimensional local analytically reduced and residually rational Cohen-Macaulay rings, cf. Th.(4.8).

In section 2 we prove some results on pseudo-valuation rings which in this form could not be found in the literature. In section 3 we give several characterizations of Gorensteinness in terms of dimension formulae for certain vector spaces associated to ideals, cf.(3.6) and (3.7). In section 4 we prove the above mentioned characterization; we have to assume that the ring is residually rational [cf.(4.1) for definition], and that the residue field is infinite. In section 5 we give another characterization of Gorensteinness, cf. Prop.(5.3), and finally, in section 6, we give some examples which show that the hypotheses [residually rational and infinite residue field] are necessary.