Finite length modules over HNPs.

Abstract

We characterize a finite length module over HNP that is annihilated by an invertible ideal. We also characterize finite length modules over HNP that have no composition factors annihilated by an invertible ideal. The two characterizations are used to prove Levy’s Theorem about the decomposition of finite length modules over HNPs. We also prove that the ring of matrices over a uniserial ring is serial by generalizing the technique of proving that the ring of matrices over a simple ring is simple. This is done by exploring the form of a one sided ideal of a matrix ring. We also characterize a uniserial Artinian ring as a local, principle ideal, Artinian ring. We use the two results to prove that the component that is annihilated by an invertible ideal in the Levy decomposition is a serial module.