Ideal semigroups of noetherian domains and Ponizovski decompositions

Let RR be an integral domain with quotient field KK and L⊃KL⊃K a finite extension field. By an RR-lattice in LL we mean a finitely generated RR-module containing a basis of LL over KK. The set of all RR-lattices is a commutative multiplicative semigroup. If RR 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 RR is a Dedekind domain and L⊃KL⊃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.