Ziegler's Indecomposability Criterion

Ziegler’s Indecomposability Criterion is used to prove that a totally transcendental, i.e., $\sum $ -pure injective, indecomposable left module over a left noetherian ring is a directed union of finitely generated indecomposable modules. The same criterion is also used to give a sufficient condition for a pure injective indecomposable module $_{R}U$ to have an indecomposable local dual $U_{R}^{\#}.$