We find a bound for the Goldie dimension of hereditary modules in terms of the cardinality of the generating sets of their quasi-injective hulls. Several consequences are deduced. In particular, it is shown that every finitely generated hereditary module with countably generated quasi-injective hull is noetherian. It is also shown that every right hereditary ring with finitely generated injective hull is right artinian, thus answering a long standing open question posed by Dung, Gómez Pardo and Wisbauer.
Dinh, Hai Q; Guilo Asensio, Pedro A; López-Permouth, Sergio R (2006). On the Goldie Dimension of Rings and Modules. Elsevier 305(2) 937-948. doi: 10.1016/J.JALGEBRA.2006.06.043. Retrieved from https://oaks.kent.edu/mathpubs/10