Modules with serial Noetherian endomorphism rings
A module is called serial if its submodules are linearly ordered with respect to inclusion. A ring \(\Lambda\) is said to be right (left) serial if \(\Lambda_{\Lambda}\) \((_{\Lambda}\Lambda)\) is a direct sum of serial submodules, and \(\Lambda\) is serial if it is both left and right serial. The paper delivers a characterization of generators with serial Noetherian endomorphism rings. Due to Warfield's structure theorem for serial Noetherian rings it suffices to obtain such a characterization separately for serial Artinian and serial prime Noetherian rings. Theorem 1. Let \(\Lambda\) be a serial Artinian ring and \(M=\oplus M_ i\) a \(\Lambda\)-module with indecomposable \(M_ i\). Then End M is serial Artinian if no \(M_ i\) is isomorphic to a proper subfactor of any \(M_ j\). Theorem 2. Let \(G_{\Lambda}\) be a generator. Then, the endomorphism ring of G is serial prime Noetherian if \(\Lambda\) is serial prime Noetherian and G is finitely generated projective.
scientific article; zbMATH DE number 4134259