  It is proven that each indecomposable injective module over a valuation
domain $R$ is polyserial if and only if each maximal immediate extension
$\widehat{R}$ of $R$ is of finite rank over the completion $\widetilde{R}$ of
$R$ in the $R$-topology. In this case, for each indecomposable injective module
$E$, the following invariants are finite and equal: its Malcev rank, its
Fleischer rank and its dual Goldie dimension. Similar results are obtained for
chain rings satisfying some additional properties. It is also shown that each
indecomposable injective module over one Krull-dimensional local Noetherian
rings has finite Malcev rank. The preservation of Goldie dimension finiteness
by localization is investigated too.
