KRULL-SCHMIDT THEOREMS IN DIMENSION-1

Let Lambda be a semiprime, module-finite algebra over a commutative no etherian ring R of Krull dimension 1. We find necessary and sufficient conditions for the Krull-Schmidt theorem to hold for all finitely gen erated Lambda-modules, and necessary and sufficient conditions for the Krull-Schmidt theorem to hold for all finitely generated torsionfree Lambda-modules (called ''Lambda-lattices'' in integral representation theory, and ''maximal Cohen-Macaulay modules'' in the dimension-one si tuation in commutative algebra).