| Abstract: | We study a one-dimensional analogue of representation-finite rings. For a left Noetherian semilocal ring R, we define an R-lattice to be a finitely generated R-module with zero socle. We call R lattice-finite if the number of isomorphism classes of indecomposable R-lattices is finite. Under this assumption, we give several equivalent criteria for the existence of Auslander–Reiten sequences in the category of R-lattices. A necessary condition is that the maximal left quotient ring of R is semisimple, and the main sufficient criterion states that R admits a semiperfect semiprime Asano left overorder. Presented by I. Reiten Mathematics Subject Classifications (2000) Primary: 16G70, 16G30; secondary: 16G60. |
