Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-23T22:58:48.188Z Has data issue: false hasContentIssue false
  • English
  • Français

ACCESSIBLE SUBRINGS AND KUROSH’S CHAINS OF ASSOCIATIVE RINGS

Part of: Rings and algebras arising under various constructions Modules, bimodules and ideals

Published online by Cambridge University Press:  18 July 2013

RYSZARD R. ANDRUSZKIEWICZ  and
MAGDALENA SOBOLEWSKA
RYSZARD R. ANDRUSZKIEWICZ*
Affiliation:
Institute of Mathematics, University of Białystok, 15-267 Białystok, Akademicka 2, Poland email [email protected]
MAGDALENA SOBOLEWSKA
Affiliation:
Institute of Mathematics, University of Białystok, 15-267 Białystok, Akademicka 2, Poland email [email protected]
*
✉[email protected]
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 article is devoted to the historical study of the ADS-problem with a special emphasis on the use of methods and techniques, emerging with the development of the theory of rings: accessible subrings, iterated maximal essential extensions of rings, completely normal rings. We construct new examples of classes for which Kurosh’s chain stabilizes at any given step. We recall the old nontrivial questions, and we pose a new one.

Keywords

stabilization of Kurosh’s chainsconstruction of classesaccessible subringsiterated maximal essential extensions of ringscompletely normal rings

MSC classification

Secondary: 16D25: Ideals 16S70: Extensions of rings by ideals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 95 , Issue 2 , October 2013 , pp. 145 - 157
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Andruszkiewicz, R. R., ‘On accessible subrings of associative rings’, Proc. Edinb. Math. Soc. 35 (1992), 101–107.CrossRefGoogle Scholar
Andruszkiewicz, R. R., ‘On iterated maximal essential extensions of rings’, Algebra Colloq. 9 (3) (2002), 241–258.Google Scholar
Andruszkiewicz, R. R., ‘The classification of integral domains in which the relation of being an ideal is transitive’, Comm. Algebra 31 (5) (2003), 2067–2093.CrossRefGoogle Scholar
Andruszkiewicz, R. R. and Puczyłowski, E. R., ‘Kurosh’s chains of associative rings’, Glasg. Math. J. 32 (1990), 67–69.CrossRefGoogle Scholar
Andruszkiewicz, R. R. and Puczyłowski, E. R., ‘Accessible subrings and Kurosh’s chains of associative rings’, Algebra Colloq. 4 (1) (1997), 79–88.Google Scholar
Andruszkiewicz, R. and Sobolewska, M., ‘On the stabilisation of one-sided Kurosh’s chains’, Bull. Aust. Math. Soc. 86 (3) (2012), 473–480.CrossRefGoogle Scholar
Armendariz, E. P. and Leavitt, W. G., ‘The hereditary property in the lower radical construction’, Canad. J. Math. 20 (1968), 474–476.CrossRefGoogle Scholar
Beidar, K. I., ‘A chain of Kurosh may have an arbitrary finite length’, Czechoslovak Math. J. 32 (107) (3) (1982), 418–422.CrossRefGoogle Scholar
Beidar, K. I., ‘Semisimple classes of algebras and the lower radical’, Mat. Issled. 105 (1988), 8–12; (in Russian).Google Scholar
Beidar, K. I., ‘On essential extensions, maximal essential extensions and iterated maximal essential extensions in radical theory’, Colloq. Math. Soc. János Bolyai, 61. Theory of Radicals, Szekszard (Hungary) (1991), 17–26.Google Scholar
Bourbaki, N., Algèbre Commutative (Hermann, Paris) 1961–1965.Google Scholar
Divinsky, N., Krempa, J. and Sulinski, A., ‘Strong radical properties of alternative and associative rings’, J. Algebra 17 (1971), 369–388.CrossRefGoogle Scholar
Filipowicz, M. and Puczyłowski, E. R., ‘When is the lower radical determined by a set of rings strong?’, Glasg. Math. J. 46 (2004), 371–378.CrossRefGoogle Scholar
Gardner, B. J. and Wiegandt, R., Radical Theory of Rings (Marcel Dekker, Inc., New York, 2004).Google Scholar
Heinicke, A. G., ‘A note on lower radical constructions for associative rings’, Canad. Math. Bull. 11 (1) (1968), 23–30.CrossRefGoogle Scholar
Hoffman, A. E. and Leavitt, W. G., ‘Properties inherited by the lower radical’, Port. Math. 27 (1968), 63–66.Google Scholar
Liu, S. X., Luo, Y. L., Tang, A. P., Xiao, J. and Guo, J. Y., ‘Some results on modules and rings’, Bull. Soc. Math. Belg. Sér. B 39 (1987), 181–193.Google Scholar
Lvov, I. V. and Sidorov, A. V., ‘On the stabilization of Kurosh’s chains’, Mat. Zametki 36 (1984), 815–821; (in Russian).Google Scholar
Puczyłowski, E. R., ‘On questions concerning strong radicals of associative rings’, Quaest. Math. 10 (1987), 321–338.CrossRefGoogle Scholar
Sands, A. D., ‘On ideals in over-rings’, Publ. Math. Debrecen 35 (1988).Google Scholar
Stewart, P. N., ‘On the lower radical construction’, Acta Math. Acad. Sci. Hungar. 25 (1–2) (1974), 31–32.CrossRefGoogle Scholar
Sulinski, A., Anderson, R. and Divinsky, N., ‘Lower radical properties for associative and alternative rings’, J. Lond. Math. Soc. 41 (1966), 417–424.CrossRefGoogle Scholar
Watters, J. F., ‘On the lower radical construction for algebras’, Preprint 1985.Google Scholar
‘Radicals of rings and related topics’, 2009, http://aragorn.pb.bialystok.pl/~piotrgr/BanachCenter/meeting.html.Google Scholar