Ideal semigroups of noetherian domains and Ponizovski decompositions

Let R be an integral domain with quotient field K and L superset of K a finite extension field. By an R-lattice in L we mean a finitely generated R-module containing a basis of L over K. The set of all R-lattices is a commutative multiplicative semigroup. If R is one-dimensional and noetherian, we determine the structure of this semigroup and of the corresponding class semigroup by means of its partial Ponizovski factors. If moreover R is a Dedekind domain and L superset of K is separable, we give criteria for the partial Ponizovski factors to be groups in terms of the different and the conductor of their endomorphism rings. (c) 2006 Elsevier B.V. All rights reserved.