Serial Rings

A module is called uniseriat if it has a unique composition series of finite length. A ring (always with 1) is called serial if its right and left free modules are direct sums of uniserial modules. Nakayama, who called these rings generalized uniserial rings, proved [21, Theorem 171 that every finitely generated module over a serial ring is a direct sum of uniserial modules. In section one we give a short conceptual proof of this result, strengthening it to arbitrary modules (Theorem 1.2). As a byproduct of the proof, we obtain a condition for a projective module over a serial ring to be injective (Theorem 1.4). More can be said about the structure of modules over a serial ring. In section two we show that the endomorphism ring of a projective module over a serial ring is a local serial ring (Corollary 2.2), and that the composition series of any uniserial module over a serial ring is periodic in a strong sense (Theorem 2.3). The section concludes with the theorem that any two simple modules over an indecomposable serial ring have the same endomorphism ring (Theorem 2.4). Serial rings occur naturally in several contexts. It has long been known that every proper factor ring of a (commutative) Dedekind domain is an artinian principal ideal ring. (Commutative or not, any artinian principal ideal ring is serial.) Although it is also true that factor rings of Dedekind prime rings are artinian principal ideal rings [23, Theorem 3.51, this fails for hereditary noetherian prime rings in general (see [8, Sec. 41). However, we prove in section three that any artinian factor ring of an hereditary ring with a flat injective envelope is serial (Theorem 3.1). In particular, every proper factor ring of an hereditary noetherian prime ring is serial. Are Dedekind prime rings precisely the hereditary noetherian prime rings whose factor rings are principal ideal rings ? The answer is “yes” under mild