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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.

Information

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

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