We prove that a Dedekind domain R, graded by a nontrivial torsionfree abelian group, is either a twisted group ring kt[G] or a polynomial ring k[X], where k is a field and G is an abelian torsionfree rank one group. It follows that R is a Dedekind domain if and only if R is a principal ideal domain. We also investigate the case when R is graded by an arbitrary nontrivial torsionfree monoid. We fix some notation and terminology. All rings R are commutative with identity 1 , and all semigroups S are torsionfree. In case S is a monoid, we denote by e the identity of S and by W(S) the group of invertible elements of S. For s E S , denote by (s) (respectively (s) I ) the subsemigroup (respectively submonoid) of S generated by s. For more details on semigroups we refer to [3]. We say that R is S-graded if R sEsRs , a direct sum of additive subgroups, such that R R1 C Rs, for all s, t E S. The set h(R) = Us sRs is the set of all homogeneous elements. If T is a subset of S, then we put R[T] teTTRt. Clearly, if Supp(R) ={s E SIRs $ O}, the support of R, then R = R[Supp(R)]' Obviously Supp(R) is a monoid if R is a domain. If, moreover, S is a group and RsRt =Rst for all s, t c S, then R is called strongly S-graded. If I is an ideal of R, we denote by (I)h the ideal generated by all homogeneous elements of I. If I = ()h , then I is called a homogeneous ideal of R. If S is a monoid and R is an S-graded integral domain, then Qg(R) {rcr c R, 0 #& c E Rs , s c S}, the graded quotient ring of R; if moreover, S is cancellative, then Qg(R) is G-graded, where G is the quotient group of 5, and its component of degree e is clearly a field. For further details on graded rings we refer to [6]. In recent years there has been a growing interest in divisibility properties of graded rings. For example, in [1, 7] graded rings which are factorial domains are investigated, while in [1, 2] graded rings that are Krull domains are studied. In this paper we investigate graded rings which are Dedekind domains. Received by the editors April 10, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 1 3F05, 1 6A03. The first author is a research assistant at the National Fund for Scientific Research, Belgium. The second author is supported in part by NSERC-grant OGP003663 1. ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page