Ziegler's indecomposability criterion. (Q2861274)

This paper has two main foci, both of which rest on an application of Ziegler's Indecomposability Criterion. First, let \(_RU\) be a pure injective indecomposable left \(R\)-module. Then \(_RU\) has a local endomorphism ring \(S=\mathrm{End}_RU\), which may be considered as an \(R\)-\(S\)-bimodule and the top of \(S\) is a division ring \(\Delta\), which may be made into a right \(S\)-module \(\Delta_S\). Now \(\Hom_S(_RU_S,E_S)\) is called the local dual of \(_RU\) and is denoted by \(U_R^\#\). Three equivalent conditions on a pure injective indecomposable left \(R\)-module \(_RU\) that force \(U_R^\#\) to be an indecomposable right \(R\)-module are found. Second, if \(R\) is a left Noetherian ring and \(M\) is a totally transcendental indecomposable left \(R\)-module then \(M\) is shown to be a directed union of finitely generated indecomposable modules. The importance of this result is illustrated by showing how it may be applied to generic modules.