Vol. 41, No. 3, 1972

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Rings of quotients of endomorphism rings of projective modules

Robert S. Cunningham, Edgar Andrews Rutter and Darrell R. Turnidge

Vol. 41 (1972), No. 3, 647–668

This paper investigates two related problems. The first is to describe the double centralizer of an arbitrary projective right R-module. This proves to be the ring of left quotients of R with respect to a certain canonical hereditary torsion class of left R-modules determined by the projective module.

The second is to determine the relationship between rings of left quotients of R and S, where S is the endomorphism ring of a finitely generated projective right R-module PR. It is shown that there exists an inclusion-preserving, one-to-one correspondence between hereditary torsion classes (or localizing subcategories) of left S-modules and hereditary torsion classes of left R-modules which contain the canonical torsion class determined by PR.

If QR and Qs are rings of left quotients with respect to corresponding classes, then P RQR is a finitely generated projective right QR-module with QS as its QR-endomorphism ring. Necessary and sufficient conditions are obtained for the maximal rings of left quotients to be related in this manner. In particular, this occurs when PR is a faithful R-module and R is either a semi-prime ring or a ring with zero left singular ideal. The situation considered includes the case where S is an arbitrary ring, SP is a left S-generator, and R is the S-endomorphism ring of SP. When SP is a projective left S-generator, the maximal rings of left quotients of R and S are related in the manner considered above.

Mathematical Subject Classification
Primary: 16A08
Received: 2 November 1970
Revised: 12 October 1971
Published: 1 June 1972
Robert S. Cunningham
Edgar Andrews Rutter
Darrell R. Turnidge