OBNER BASES AND PRIMARY DECOMPOSITION IN POLYNOMIAL RINGS IN ONE VARIABLE Summary: GR š OBNER BASES AND PRIMARY DECOMPOSITION IN POLYNOMIAL RINGS IN ONE VARIABLE OVER DEDEKIND DOMAINS W.W. ADAMS AND P. LOUSTAUNAU Abstract. Let D be a Dedekind domain with quotient field K, let x be a single variable, and let I be an ideal in D[x]. In this paper we will describe explicitly the structure of a Gršobner basis for I and we will use this Gršobner basis to compute the primary decomposition of I . This Gršobner basis also has a property similar to that of strong Gršobner bases over PID's ([7], see also [1]). 1. Introduction Let D be a Dedekind domain with quotient field K and let x be a single variable. Let I be an ideal in D[x]. The main result inn Section 2 is a structure theorem for a special Gršobner basis for I. First, in Proposition 2.2, we factor out the greatest common divisor of I which, by Corollary 2.8, reduces the problem to the case of ideals J such that J `` D 6= f0g. In this case we show, in Theorem 2.4, that J has a Gršobner basis of the form G = fa 1 ; a 2 h 2 ; : : : ; a t h t g; where a 1 ( a 2 ( \Delta \Delta \Delta ( a t are ideals in D, h 2 ; : : : ; h t are monic polynomials in D[x] of increasing degree satisfying one additional condition. This structure theorem is similar to the one of Szekeres [10] and Lazard [7] in the case where D is a PID (see also [1]). In Corollary 2.5 Collections: Mathematics