| Popis: |
Let Λ be a module-finite algebra over a commutative noetherian ring of Krull dimension 1. We extend Roiter's direct-summand theorem to arbitrary finitely generated Λ-modules, obtaining a sharpened form of Serre's direct-summand theorem in this setting. We also extend Drozd's cancellation theorem to arbitrary finitely generated Λ-modules, obtaining a sharpened form of Bass's cancellation theorem in this setting. A corollary is that, over commutative reduced noetherian rings of dimension 1, direct-sum cancellation holds in every genus of finitely generated modules. (This becomes false if the ring has nilpotent elements.) Another corollary is that if direct-sum cancellation holds in the genera of Λ-modules M and N, then it holds in the genus of M ⊕ N. This seems to be new, even for the modules that occur in integral representation theory. The main thrust of this paper is to close the gap between integral representation theory and the rest of module theory by eliminating hypotheses concerning the existence of maximal orders (of “finite normalization,” in the commutative case) and allowing our rings to have nilpotent ideals. |
