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Locally irreducible rings

Published online by Cambridge University Press:  17 April 2009

C. Vinsonhaler  and
W. Wickless
C. Vinsonhaler
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Conn. 06268, U.S.A.
W. Wickless
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Conn. 06268, U.S.A.
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Abstract

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In the study of torsion-free abelian groups of finite rank the notions of irreducibility, field of definition and E-ring have played significant rôles. These notions are tied together in the following theorem of R. S. Pierce:

THEOREM. Let R be a ring whose additive group is torsion free finite rank irreducible and let Γ be the centralizer of QR as a QE(R) module. Then Γ is the unique smallest field of definition of R. Moreover, Γ ∩ R is an E-ring, in fact, it is a maximal E-subring of R.

In this paper we consider extensions of Pierce's result to the infinite rank case. This leads to the concept of local irreducibility for torsion free groups.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 32 , Issue 1 , August 1985 , pp. 129 - 145
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Arnold, D. M., “Strongly homogeneous torsion-free groups of finite rank”, Proc. Amer. Math. Soc. 56 (1976), 67–72.CrossRefGoogle Scholar
[2]Arnold, D. M., Pierce, R. S., Reid, J. D., Vinsonhaler, C. I., Wickless, W. J., “Torsion-free abelian groups of finite rank projective as modules over their endomorphism rings”, J. Algebra 71 (1981), 1–10.CrossRefGoogle Scholar
[3]Beaumont, R. A. and Pierce, R. S., “Torsion-free rings”, Illinois J. Math. 5 (1961), 61–98.CrossRefGoogle Scholar
[4]Bowshell, R. and Schultz, P., “Unital rings whose additive endomorphisins commute”, Math. Ann. 228 (1977), 197–214.CrossRefGoogle Scholar
[5]Krylov, P. A., “Strongly homogeneous torsion-free abelian groups”, Siberian Math. J. 24 (1983), 77–84.CrossRefGoogle Scholar
[6]Lady, L., “A seminar on splitting rings for torsion-free modules over Dedekind domains”, Lecture Notes in Mathematics 1006 Springer-Verlag (1983), 1–49.Google Scholar
[7]Pierce, R. S., “Subrings of simple algebras”, Michigan Math. J. 7 (1960), 241–243.CrossRefGoogle Scholar
[8]Pierce, R. S., “Associative Algebras”, Graduate Texts in Mathematics, Springer-Verlag 88 (1982).Google Scholar
[9]Pierce, R. S. and Vinsonhaler, C., “Realizing central division algebras”, Pacific J. Math. 109 (1983), 165–177.CrossRefGoogle Scholar
[10]Reid, J. D., “On the ring of quasi-endomorphisms of a torsion-free groups”, Topics in Abelian Groups, Chicago, 1963, 51–68.Google Scholar
[11]Reid, J. D., “On rings on groups”, Pacific J. Math. 53 (1974), 229–237.CrossRefGoogle Scholar
[12]Reid, J. D., “Abelian groups finitely generated over their endomorphism rings” Lecture Notes in Mathematics 874 Springer-Verlag (1981), 41–52.Google Scholar
[13]Reiner, I., “Maximal Orders”, London Math. Soc. Monographs, Academic Press (1975).Google Scholar
[14]Schultz, P., “The endomorphism ring of the additive group of a ring”, J. Austral. Math. Soc. 15 (1973), 60–69.CrossRefGoogle Scholar