Serial Krull–Schmidt rings and pure-injective modules

A ring is called Krull–Schmidt if every finitely presented module over it can be decomposed into direct sum of modules with local endomorphism rings. The serial Krull–Schmidt rings are described as rings with the weak invariance condition. The classification of indecomposable pure-injective modules over uniserial ring is simplified and criteria for the existence of superdecomposable pure-injective module is given for semi-invariant case. Let $T$ be the theory of all modules over effectively given invariant uniserial ring $R$ with infinite residue skew field. It is shown that $T$ is decidable if the question of invertibility of element from $R$ can be solved effectively.