Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T08:14:10.691Z Has data issue: false hasContentIssue false
  • English
  • Français

Endomorphism rings of Butler groups

Part of: Abelian groups

Published online by Cambridge University Press:  09 April 2009

D. M. Arnold  and
C. I. Vinsonhaler
D. M. Arnold
Affiliation:
Department of Mathematical SciencesNew Mexico State UniversityLas Cruces, New Mexico 88003, U.S.A.
C. I. Vinsonhaler
Affiliation:
Department of MathematicsUniversity of ConnecticutStorrs, Connecticut 06268, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This note is devoted to the question of deciding whether or not a subring of a finite-dimensional algebra over the rationals, with additive group a Butler group, is the endomorphism ring of a Butler group (a Butler group is a pure subgroup of a finite direct sum of rank-1 torsion-free abelian groups). A complete answer is given for subrings of division algebras. Several applications are included.

MSC classification

Secondary: 20K15: Torsion-free groups, finite rank
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 42 , Issue 3 , June 1987 , pp. 322 - 329
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Arnold, D., Finite rank torsion-free abelian groups and rings, Lecture Notes in Mathematics 931, (Springer-Verlag, 1982).CrossRefGoogle Scholar
[2]Arnold, D., ‘Pure subgroups of finite rank completely decomposable groups’, Proceedings of Oberwolfach abelian group theory conference, Lecture Notes in Mathematics 874, (Springer-Verlag, 1981, pp. 1–31).Google Scholar
[3]Arnold, D. and Vinsonhaler, C., ‘Pure subgroups of finite rank completely decomposable groups II’, Proceedings of Hawaii abelian group theory conference, Lecture Notes in Mathematics 1006, (Springer-Verlag, 1983, pp. 97–143).CrossRefGoogle Scholar
[4]Brenner, S., ‘Endomorphism algebras of vector spaces with distinguished sets of subspaces’, J. Algebra 6 (1967), 100–114.CrossRefGoogle Scholar
[5]Brenner, S. and Butler, M. C. R., ‘Endomorphism rings of vector spaces and torsion free abelian groups’, J. London Math. Soc. 40 (1965), 183–187.CrossRefGoogle Scholar
[6]Butler, M. C. R., ‘A class of torsion-free abelian groups of finite rank’, Proc. London Math. Soc. 15 (1965), 680–698.CrossRefGoogle Scholar
[7]Butler, M. C. R., ‘Torsion-free modules and diagrams of vector spaces’, Proc. London Math. Soc. 18 (1968), 635–652.CrossRefGoogle Scholar
[8]Corner, A. L. S., ‘Every countable reduced torsion-free ring is an endomorphism ring’, Proc. London Math. Soc. 13 (1963), 687–710.CrossRefGoogle Scholar
[9]Lady, E. L., ‘Extension of scalars for torsion free modules over Dedekind domains’, Symposia Mathematica 23 (1979), 287–305.Google Scholar
[10]Koehler, J., ‘The type set of a torsion-free group of finite rank’, Illinois J. Math. 9 (1965), 66–86.CrossRefGoogle Scholar