Krull-Schmidt theorems in dimension 1

Let $\Lambda$ be a semiprime, module-finite algebra over a commutative noetherian ring $R$ of Krull dimension 1. We find necessary and sufficient conditions for the Krull-Schmidt theorem to hold for all finitely generated $\Lambda$-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 situation in commutative algebra).