Endomorphism Rings and Gabriel Topologies

Khuri, Soumaya Makdissi

Canadian journal of mathematics, Tome 36 (1984) no. 2, pp. 193-205

Résumé

A basic tool in the usual presentation of the Morita theorems is the correspondence theorem for projective modules. Let RM be a left R-module and B = HomR (M, M). When M is a progenerator, there is a close connection (in fact a lattice isomorphism) between left R-submodules of M and left ideals of B, which can be applied to the solution of problems such as characterizing when the endomorphism ring of a finitely generated projective faithful module is simple or right Noetherian. More generally, Faith proved that this connection can be retained in suitably modified form when M is just a generator in R-mod([4], [2], [3]). In this form the correspondence theorem can be applied to show, e.g., that, when RM is a generator, then (a): RM is finite-dimensional if and only if B is a left finite-dimensional ring and in this case d(R M) = d(B B), and (b): If RM is nonsingular then B is a left nonsingular ring ([6]).

Détail

DOI : 10.4153/CJM-1984-013-4

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