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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.
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