Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-15T01:22:48.762Z Has data issue: false hasContentIssue false
  • English
  • Français

The characterization problem for endomorphism rings

Published online by Cambridge University Press:  09 April 2009

J. L. García
J. L. García
Affiliation:
Universidad de Murcia30001 Murcia, Spain
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 consider the problem of characterizing by abstract properties the rings which are isomorphic to the endomorphism ring End (RF) of some free module F over a ring R in a given class R of rings. We solve this problem when R is any class of rings (by employing topological notions) and when R is the class of all the left Kasch rings (in terms of algebraic properties only).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 1 , February 1991 , pp. 116 - 137
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Anderson, F. W. and Fuller, K. R., Rings and categories of modules, (Springer-Verlag, New York, 1973).Google Scholar
[2]Brodskii, G. M., ‘Annihilator conditions in endomorphism rings of modules’, Math. Notes 16 (1974), 1153–1158.Google Scholar
[3]Franzsen, W. N. and Schultz, P., ‘The endomorphism ring of a locally free module’, J. Austral. Math. Soc. Ser. A 35 (1983), 308–326.Google Scholar
[4]Fuller, K. R., ‘Density and equivalence’, J. Algebra 29 (1974), 528–550.Google Scholar
[5]Fuchs, L., Infinite abelian groups, Vol. 1 (Academic Press, New York, 1970).Google Scholar
[6]García, J. L., ‘Morita-like equivalences of categories of modules’, preprint.Google Scholar
[7]García, J. L., ‘The finite column matrix ring of a ring’, Proceedings of the First Spanish-Belgian Week on Algebra and Geometry, eds. Bueso, J. L., Segura, M. I., Verschoren, A., R. U. C. A., Antwerpen, 1988, pp. 64–74.Google Scholar
[8]García, J. L. and Pardo, J. L. Gómez, ‘On endomorphism rings of quasi-projective modules’, Math. Z. 196 (1987), 87–108.Google Scholar
[9]García, J. L. and Saorín, M., ‘Endomorphism rings and category equivalences’, J. Algebra, 127 (1989), 324–350.Google Scholar
[10]Jacobson, N., Structure of rings, (AMS Colloq. Publ., Vol. 37, Amer. Math. Soc., Providence, R.I., 1968).Google Scholar
[11]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.Google Scholar
[12]Liebert, W., ‘Endomorphism rings of reduced torsion-free modules over complete discrete valuation rings’, Trans. Amer. Math. Soc. 169 (1972), 347–363.Google Scholar
[13]Liebert, W., ‘Endomorphism rings of free modules over principal ideal domains’, Duke Math. J. 41 (1974), 323–328.Google Scholar
[14]Metelli, C. and Salce, L., ‘The endomorphism ring of an abelian torsion-free homogeneous separable group’, Arch. Math. (Basel) 26 (1975), 480–485.Google Scholar
[15]Müller, B. J., ‘The quotient category of a Morita context’, J. Algebra 28 (1974), 389–407.Google Scholar
[16]Osofsky, B. L., ‘Some properties of rings reflected in endomorphism rings of free modules’, in Contemporary Math., Vol. 13, pp. 179–181, (Amer. Math. Soc., Providence, R.I., 1982).Google Scholar
[17]Rowen, L. H., Ring theory, Vol. I, (Academic Press, Boston, Mass., 1988).Google Scholar
[18]Stenström, B., Rings of quotients, (Springer-Verlag, Berlin, 1975).Google Scholar
[19]Tominaga, H., ‘On s-unital rings’, Math. J. Okayama Univ. 18 (1976), 117–134.Google Scholar
[20]Wolfson, K. G., ‘An ideal-theoretic characterization of the ring of all linear transformations’, Amer. J. Math. 75 (1953), 358–386.Google Scholar
[21]Zimmermann, B., ‘Endomorphism rings of self-generators’, Algebra Berichte 27, Math. Inst. Univ. München (1975).Google Scholar