Generalized rees rings and arithmetical graded rings

Abstract

0. INTRODUCTION In this paper we study graded rings with an arithmetical ideal theory for the graded ideals, e.g., Gr-Dedekind and Gr-principal ideal rings. If these rings are positively graded rings, then the structure of Gr-Dedekind and of Gr-principal ideal rings is easily investigated and it is much like the structure of the ungraded equivalents. For arbitrary Z-gradations, however, the new classes of rings introduced here have an interesting structure relating to the class group of the part of degree zero. The main results in Section 2 determine the structure of Gr-Dedekind rings. First, if R is a Gr-Dedekind ring such that RR, = R, then R is a generalised Rees ring (and vice versa). These are obtained as follows: let I be a fractional ideal of a Dedekind domain R,, consider the graded ring l?,(I) = CnCZ I”X” which is a graded subring of K,[X, X-I], K, being the field of fractions of R,. Nate that classically, the Rees ring of an ideal I of a domain R was defined to be the graded ring R(~)=ROIO...OI”O...rR+I;Y+I’X*+...+ p/y* + . . . . Now, for an arbitrary Gr-Dedekind ring R, the part of degree 0, R, say, is a Dedekind ring and in Theorem 2.10 we establish that there is an e E n\l such that R(‘) is a generalized Rees ring, where R(‘) is the graded ring defined by (R @J)k = R,,. The closing paragraphs of Section 2 deal with the study of the class groups of Gr-Dedekind rings R and the reiations between these and the class groups of R, the part of degree 0. In particular, we pay attention to some connections between the structure of class groups and the ramification in R of prime ideals of R,, containing a certain ideal 6(R), called the discriminator of R, i.e., 6(R) = R _, R i. In [5 1, we give some applications of these ideas, in particular the constructions of a “graded” Zeta-function etc... _ Another application is to the theory of the Brauer group of a commutative ring, where arithmetical graded rings play a very peculiar role. We hope ‘ti present this material in a forthcoming paper. ! 8.5