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Categorical compactness for rings

Part of: General theory of categories and functors

Published online by Cambridge University Press:  09 April 2009

Temple H. Fay  and
Stephan V. Joubert
Temple H. Fay
Affiliation:
The University of Southern MississippiHattiesburg, MS, USA
Stephan V. Joubert
Affiliation:
The Technikon Pretoria Pretoria West 0001, South Africa
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Abstract

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In this paper we study categorical compactness with respect to a class of objects F being motiveated by examples arising from modules, abelian groups, and various classes of non-abelian groups. This theory is then applied to the category of not necessarily associative rings. In particular, we study the example arising from the class of all torsion-free rings. This work extends some recent results of B. J. Gardner for associative rings and radical classes.

Keywords

torsion theorysemisimplepolynomial regularityclosure operatorcategorical compactness.

MSC classification

Secondary: 18A20: Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 59 , Issue 3 , December 1995 , pp. 313 - 329
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Cassidy, C., Hébert, M. and Kelly, G. M., ‘Reflective subcategories, localizations and factorization systems’, J. Austral. Math. Soc. (Series A) 38 (1985), 287–329.Google Scholar
[2]Castellini, G., ‘Compact objects, surjectivity of epimorphisms and compactifications’, Cahiers Topologie Géom. Différentielle Catégoriques XXXI-1 (1990), 53–65.Google Scholar
[3]Dikranjan, D. and Giuli, E., ‘Closure operators I’, Topology Appl. 27 (1987), 129–143.Google Scholar
[4]Dikranjan, D., ‘Factorizations, injectivity and compactness in categories of modules’, Comm. Algebra 19 (1991), 45–83.CrossRefGoogle Scholar
[5]Fay, T. H., ‘Compact modules’, Comm. Algebra 16 (1988), 1209–1219.Google Scholar
[6]Fay, T. H., ‘Remarks on the Mal'cev completion of torsion-free locally nilpotent groups’, Cahiers Topologie Géom. Différentielle Catégoriques XXXV-1 (1994), 75–84.Google Scholar
[7]Fay, T. H. and Schoeman, M. J., ‘The torsion completion, the p-adic completion, and categorical compactness for nilpotent groups’, in preparation.Google Scholar
[8]Fay, T. H. and Walls, G. L., ‘A characterization of categorically compact locally nilpotent groups’, Comm. Algebra, to appear.Google Scholar
[9]Fay, T. H., ‘Completions and categorical compactness for nilpotent groups’, Quaestiones Math., to appear.Google Scholar
[10]Fuchs, L., Infinite abelian groups I, II (Academic Press, New York, 1973).Google Scholar
[11]Gardner, B. J., ‘Radicals which define factorization systems’, Comment. Math. Univ. Carolin. 32 (1991), 601–607.Google Scholar
[12]Golan, J. S., Torsion theories, Pitman Monographs and Surveys in Pure and Applied Math. (Longman Scientific and Technical, Harlow, 1986).Google Scholar
[13]Herrlich, H., Salicrup, G. and Strecker, G. E., ‘Factorizations, denseness, separation, and relatively compact objects’, Topology Appl. 27 (1987), 157–169.Google Scholar
[14]Hoehnke, H-J., ‘Radikale in allegemeinen algebren’, Math. Nachr. 32 (1966), 347–383.Google Scholar
[15]Joubert, S. V., ‘Categorically compact rings and modules’, Chinese J. Math. 20 (1992), 347–365.Google Scholar
[16]Leavitt, W. G. and Wiegandt, R., ‘Torsion theory for not necessarily associative rings’, Rocky Mountain J. Math. 9 (1979), 259–271.Google Scholar
[17]Manes, E. G., ‘Compact Hausdorff objects’, Topology Appl. 4 (1979), 341–360.Google Scholar
[18]Roos, C., ‘A class of regularities for rings’, J. Austral. Math. Soc. (Series A) 27 (1979), 437–453.CrossRefGoogle Scholar