  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 answering a long standing open question posed by Dung, Gomez
Pardo and Wisbauer.
