Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-06-04T16:17:37.627Z Has data issue: false hasContentIssue false
  • English
  • Français

The endomorphism ring of a locally free module

Published online by Cambridge University Press:  09 April 2009

W. N. Franzsen  and
P. Schultz
W. N. Franzsen
Affiliation:
Department of MathematicsUniversity of Western AustraliaNedlands, W. A. 6009, Australia
P. Schultz
Affiliation:
Department of MathematicsUniversity of Western AustraliaNedlands, W. A. 6009, Australia
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.

We identify a large class of rings over which locally free modules are determined by their endomorphism rings. We characterize these endomorphism rings and consider under what circumstances the conditions on the locally free modules can be relaxed, for example by requiring that only one of the rings need be in the special class, or by replacing ‘free' by “projective”.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 35 , Issue 3 , December 1983 , pp. 308 - 326
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Baer, R., Linear algebra and projective geometry (Academic Press, New York, 1952).Google Scholar
[2]Bazzoni, S. and Metelli, C., ‘On abelian torsion-free separable groups and their endomorphism rings’, Symposia Math. 23 (1979), 259–285.Google Scholar
[3]Cohn, P. M., ‘Some remarks on the invariant basis property’, Topology 5 (1966), 215–228.CrossRefGoogle Scholar
[4]Cohn, P. M., Algebra, Vols. 1 and 2 (Wiley, New York, 1974 and 1977).Google Scholar
[5]Fuchs, L., Infinite abelian groups, Vols. 1 and 2 (Academic Press, New York, 1970 and 1973).Google Scholar
[6]Fuchs, L. and Schultz, P., ‘Endomorphism rings of valued vector spaces’, Rend. Sem. Mat. Univ. Padova 65 (1981).Google Scholar
[7]Hauptfleisch, C. J., ‘Torsion-free abelian groups with isomorphic endomorphism rings’, Arch. Math. (Basel) 24 (1973), 269–273.CrossRefGoogle Scholar
[8]Jacobson, N., The theory of rings (Math. Surveys 2, Amer. Math. Soc., Providence, R. I., 1943).CrossRefGoogle Scholar
[9]Jacobson, N., Structure of rings (Amer. Math. Soc. Colloq. Publ., 37, 1956).Google Scholar
[10]Kaplansky, I., Infinite abelian groups (Univ. of Michigan, 1954).Google Scholar
[11]Leavitt, W. G., ‘The module type of a ring’, Trans. Amer. Math. Soc. 103 (1962), 113–130.CrossRefGoogle Scholar
[12]Liebert, W., ‘Charakterisierung der Endomorphismenringe beschränkter abelscher Gruppen’, Math. Ann. 174 (1967), 217–232.CrossRefGoogle Scholar
[13]Liebert, W., ‘Charakterisierung der Endomorphismenringe endlicher abelscher Gruppen’, Arch. Math. (Basel) 18 (1967), 128–135.CrossRefGoogle Scholar
[14]Liebert, W., ‘Endomorphism rings of abelian p-groups’, Studies on abelian groups, pp. 239–258 (Dunod, Paris and Springer-Verlag, Berlin, 1967).Google Scholar
[15]Liebert, W., ‘Characterization of the endomorphism rings of divisible torsion modules and reduced complete torsion-free modules over complete discrete valuation rings’, Pacific J. Math. 37 (1971), 141–170.CrossRefGoogle Scholar
[16]Liebert, W., ‘Endomorphism rings of reduced complete torsion-free modules over complete discrete valuation rings’, Proc. Amer. Math. Soc. 36 (1972), 375–378.CrossRefGoogle Scholar
[17]Liebert, W., ‘Endomorphism rings of reduced torsion-free modules over complete discrete valuation rings’, Trans. Amer. Math. Soc. 169 (1972), 347–363.CrossRefGoogle Scholar
[18]Liebert, W., ‘Endomorphism rings of free modules over principal ideal domains’, Duke Math. J. 41 (1974), 323–328.CrossRefGoogle Scholar
[19]May, W. and Toubassi, E., ‘Endomorphisms of Abelian groups and the theorem of Baer and Kaplansky’, J. Algebra 43 (1976), 1–13.CrossRefGoogle Scholar
[20]May, W. and Toubassi, E., ‘Isomorphisms of endomorphism rings of rank one mixed groups’, J. Algebra 11 (1981), 508–514.CrossRefGoogle Scholar
[21]Metelli, C. and Salce, L., ‘The endomorphism ring of an abelian torsion-free homogeneous separable group’, Arch. Math. (Basel) 26 (1975), 480–485.CrossRefGoogle Scholar
[22]Pierce, R. S., ‘Endomorphism rings of primary abelian groups’, Proc. Colloq. on Abelian Groups, Tihany, Budapest, 1964, pp. 125–137.Google Scholar
[23]Simis, A., When are projective modules free? (Queen's Papers in Pure and Applied Mathematics, 21, 1969).Google Scholar
[24]Webb, M. C., ‘The endomorphism ring of pointed separable torsion-free abelian groups’, J. Algebra 55 (1978), 446–454.CrossRefGoogle Scholar
[25]Wolfson, K. G., ‘An ideal theoretic characterization of the ring of all linear transformations’, Amer. J. Math. 75 (1953), 358–386.CrossRefGoogle Scholar
[26]Wolfson, K. G., ‘Isomorphisms of the endomorphism rings of torsion-free modules’, Proc. Amer. Math. Soc. 13 (1962), 712–714.CrossRefGoogle Scholar
[27]Wolfson, K. G., ‘Isomorphisms of the endomorphism ring of a free module over a principal left ideal domain’, Michigan Math. J. 9 (1962), 69–75.CrossRefGoogle Scholar
[28]Wolfson, K. G., ‘Isomorphisms of the endomorphism rings of a class of torsion-free modules’, Proc. Amer. Math. Soc. 14 (1963), 589–594.CrossRefGoogle Scholar