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Examples of chain domains

Authors: R. Mazurek and E. Roszkowska
Journal: Proc. Amer. Math. Soc. 126 (1998), 661-667
MSC (1991): Primary 16D15, 16D25; Secondary 16N80
MathSciNet review: 1423319
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Abstract: Let $\gamma $ be a nonzero ordinal such that $\alpha +\gamma =\gamma $ for every ordinal $\alpha <\gamma $. A chain domain $R$ (i.e. a domain with linearly ordered lattices of left ideals and right ideals) is constructed such that $R$ is isomorphic with all its nonzero factor-rings and $\gamma $ is the ordinal type of the set of proper ideals of $R$. The construction provides answers to some open questions.

References [Enhancements On Off] (What's this?)

  • 1. U. Albrecht and G. Törner, Group rings and generalized valuations, Comm. Algebra 12 (1984), 2243-2272. MR 85f:16013
  • 2. V. A. Andrunakievich and J. M. Ryabukhin, Radicals of algebras and structural theory (Russian), Nauka, Moscow, 1979. MR 82a:16001
  • 3. P. M. Cohn, Free rings and their relations, London Math. Soc. Monographs No. 19, Academic Press, London, 1985. MR 87e:16006
  • 4. N. I. Dubrovin, Chain domains (Russian), Vestnik Moscov. Univ. Ser. I Mat. Meh. 1980, no. 2, 51-54. MR 81g:16004
  • 5. B. J. Gardner, Simple rings whose lower radicals are atoms, Acta Math. Hungar. 43 (1984), 131-135. MR 85a:16007
  • 6. W. G. Leavitt and L. C. A. van Leeuwen, Rings isomorphic with all proper factor-rings, Ring theory (Proc. 1978 Antwerp Conf.), Marcel Dekker, New York and Basel, 1979, 783-798. MR 81i:16010
  • 7. E. R. Puczy{\l}owski, Some questions concerning radicals of associative rings, Theory of Radicals (Proc. Conf. Szekszard, 1991), 209-227, Colloq. Math. Soc. János Bolyai, Vol. 61, North-Holland, Amsterdam, 1993. MR 94j:16033
  • 8. E. R. Puczy{\l}owski and E. Roszkowska, Atoms of lattices of radicals of associative rings, Radical Theory (Proc. Conf. Sendai, 1988), 123-134. MR 90e:16009
  • 9. -, On atoms and coatoms in lattices of radicals of associative rings, Comm. Algebra 20 (1992), 955-977. MR 93e:16032
  • 10. R. L. Snider, Lattices of radicals, Pacific J. Math. 40 (1972), 207-220. MR 46:7290
  • 11. J. F. Watters, Noncommutative minimally non-Noetherian rings, Math. Scand. 40 (1977), 176-182. MR 56:15694

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Additional Information

R. Mazurek
Affiliation: Institute of Mathematics, University of Warsaw, Białystok Division, Akademicka 2, 15-267 Białystok, Poland

E. Roszkowska
Affiliation: Faculty of Economy, University of Warsaw, Białystok Division, Sosnowa 62, 15-887 Białystok, Poland
Address at time of publication: Faculty of Economy, University in Białystok, Warszawska 63, 15-062 Białystok, Poland

Received by editor(s): December 1, 1995
Received by editor(s) in revised form: August 27, 1996
Communicated by: Ken Goodearl
Article copyright: © Copyright 1998 American Mathematical Society

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