Let R be a ring with 1 and consider the problem of assigning numerical invariants to right K-modules so that two such modules are elementarily equivalent if and only if they have the same invariants. This general problem has been solved by Baur [1], Garavaglia [8], and Monk [18]. The invariants they produce are however rather more model-theoretic than algebraic and are in general highly redunda...

We find a bound for the Goldie dimension of hereditary modules in terms of the cardinality of the generator sets of its quasi-injective hull. Several consequences are deduced. In particular, it is shown that every right hereditary module with countably generated quasi-injective hull is noetherian. Or that every right hereditary ring with finitely generated injective hull is artinian, thus answe...