On the Goldie Dimension of Hereditary Rings and Modules
  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. 
H.Q. Dinh, P.A. Guil Asensio and S.R. Lopez-Permouth