Regular over-orders of lattice-finite rings and the Krull–Schmidt property

Let R be a lattice-finite noetheriansemilocal ring without simple left ideals. In Rump (Preprint) we prove that R is an order in a semisimple ring Q. We refine this result by showing that R has a semiperfect regular over-order if the category R-lat of R-lattices has the Krull–Schmidt property. Together with the results of Rump this implies that both conditions are equivalent to the existence of almost split sequences in R-lat.