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MODULES WITH ABELIAN ENDOMORPHISM RINGS

Part of: Modules, bimodules and ideals

Published online by Cambridge University Press:  18 June 2010

GRIGORE CĂLUGĂREANU  and
PHILL SCHULTZ
GRIGORE CĂLUGĂREANU
Affiliation:
Babeş-Bolyai University Cluj-Napoca, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-400084 Cluj-Napoca, Romania (email: [email protected])
PHILL SCHULTZ*
Affiliation:
School of Mathematics and Statistics, University of Western Australia, Nedlands, WA 6009, Australia (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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The results of Szele and Szendrei [‘On Abelian groups with commutative endomorphism rings’, Acta Math. Acad. Sci. Hungar.2 (1951), 309–324] characterizing abelian groups with commutative endomorphism rings are generalized to modules whose endomorphism rings have various restrictions on their idempotents. Such properties include central or commuting idempotents, and one-sided ideals being two-sided. Related properties include direct summands having unique complements, or being fully invariant.

Keywords

commutative endomorphism ringscentral or commuting idempotentsduo ringsunique complementsfully invariant submodules

MSC classification

Secondary: 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 1 , August 2010 , pp. 99 - 112
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Abraham, R. and Schultz, P., ‘Additive Galois theory of modules’, in: Rings, Modules, Algebras and Abelian Groups, Lecture Notes in Pure and Applied Mathematics, 236 (Marcel Dekker, New York, 2004), pp. 1–12.Google Scholar
[2]Birkenmeier, G. F., ‘Idempotents and completely semiprime ideals’, Comm. Algebra 11 (1983), 567–580.CrossRefGoogle Scholar
[3]Camillo, V. and Nielsen, P. P., ‘McCoy rings and zero-divisors’, J. Pure Appl. Algebra 212 (2008), 599–615.CrossRefGoogle Scholar
[4]Fuchs, L., Infinite Abelian Groups, Vols. I and II (Academic Press, New York, 1970), 1973.Google Scholar
[5]Krylov, P. A., Mikhalev, A. V. and Tuganbaev, A. A., Endomorphism Rings of Abelian Groups, Algebras and Applications, 2 (Kluwer Academic Publishers, Dordrecht, 2003).CrossRefGoogle Scholar
[6]Lam, T. Y., Exercises in Classical Ring Theory, Problem Books in Mathematics (Springer, New York, 1995).CrossRefGoogle Scholar
[7]Reid, J. D., ‘On subcommutative rings’, Acta Math, Acad. Sci. Hungar. 16 (1965), 23–26.CrossRefGoogle Scholar
[8]Schultz, P., ‘The endomorphism ring of the additive group of a ring’, J. Aust. Math. Soc. 15(1) (1973), 60–69.CrossRefGoogle Scholar
[9]Szele, T. and Szendrei, J., ‘On Abelian groups with commutative endomorphism rings’, Acta Math. Acad. Sci. Hungar. 2 (1951), 309–324.CrossRefGoogle Scholar