On the Goldie dimension of rings and modules

Abstract

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.

Modul Herediter atas aljabar Lintasan Leavitt dari Graf A∞

“…Modul herediter merupakan perumuman dari gelanggang herediter, pertama kali diperkenalkan oleh Shrikhande [10] pada tahun 1973. Studi tentang modul herediter atas gelanggang telah banyak dilakukan seperti dapat dilihat pada [5,7].…”

Hereditary rings with countably generated cotorsion envelope

“…We consider the special case in which the given ring R is left hereditary and we try to find conditions on its cotorsion envelope which ensure that R is semilocal. These conditions are inspired by [5]. In that paper, authors showed that any left hereditary ring having a countably generated left injective envelope is left noetherian.…”

“…In particular, we deduce that left hereditary rings with finitely generated left cotorsion envelope are just the left hereditary left cotorsion semiperfect rings, thus obtaining the structure of them. We would like to stress that our proof relies on set theoretical arguments which have their origin in the decomposition of infinite sets into almost disjoint subsets obtained by Tarski in [18,Théorème 7] and that were also used in [5,15] to obtain their main results. However, the situation here is much more difficult to handle since we do not have unique cotorsion envelopes of pure submodules of C = C ( R R) within C .…”

“…However, the situation here is much more difficult to handle since we do not have unique cotorsion envelopes of pure submodules of C = C ( R R) within C . This fact was essential in [5,15], where uniqueness of injective envelopes is a key fact (see [16] for a discussion on this question). Indeed, our result shows one of the first situations in which this kind of techniques is successfully applied in absence of uniqueness of envelopes.…”