ABSTRACT
A module M is said to be super-decomposable if M does not contain indecomposable direct summands.
The existence of a super-decomposable pure-injective module over a given ring R is a natural question in the theory of modules. The answer depends on a variety of ranks defined on the lattice of all pp-formulae over R and may be considered as a measure of complexity of the entire category of .R-modules. For instance, according to Trlifaj [10], over a von Neumann regular ring R, a super-decomposable (pure) injective module exists iff R is not semiartinian. Rather few cases are known where such a satisfactory answer can be obtained. For example Puninski (see [7, Ch. 12]) proved that over a commutative valuation ring V, a super-decomposable pureinjective module exists iff the Krull dimension of V is undefined.
