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Commutative Endomorphism Rings

Published online by Cambridge University Press:  20 November 2018

J. Zelmanowitz
J. Zelmanowitz*
Affiliation:
University of California, Los Angeles, California
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The problem of classifying the torsion-free abelian groups with commutative endomorphism rings appears as Fuchs’ problems in [4, Problems 46 and 47]. They are far from solved, and the obstacles to a solution appear formidable (see [4; 5]). It is, however, easy to see that the only dualizable abelian group with a commutative endomorphism ring is the infinite cyclic group. (An R-module Miscalled dualizable if HomR(M, R) ≠ 0.) Motivated by this, we study the class of prime rings R which possess a dualizable module M with a commutative endomorphism ring. A characterization of such rings is obtained in § 6, which as would be expected, places stringent restrictions on the ring and the module.

Throughout we will write homomorphisms of modules on the side opposite to the scalar action. Rings will not be assumed to contain identity elements unless otherwise indicated.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 1 , February 1971 , pp. 69 - 76
Copyright
Copyright © Canadian Mathematical Society 1971

References

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