Examples of chain domains

Mazurek, R.; Roszkowska, E.

doi:10.1090/S0002-9939-98-04127-6

Examples of chain domains
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by R. Mazurek and E. Roszkowska PDF

Proc. Amer. Math. Soc. 126 (1998), 661-667 Request permission

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

Ulrich Albrecht and Günter Törner, Group rings and generalized valuations, Comm. Algebra 12 (1984), no. 17-18, 2243–2272. MR 747226, DOI 10.1080/00927878408823106
V. A. Andrunakievič and Ju. M. Rjabuhin, Radikaly algebr i strukturnaya teoriya, Sovremennaya Algebra. [Modern Algebra], “Nauka”, Moscow, 1979 (Russian). MR 548864
P. M. Cohn, Free rings and their relations, 2nd ed., London Mathematical Society Monographs, vol. 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. MR 800091
N. I. Dubrovin, Chain domains, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2 (1980), 51–54, 104 (Russian, with English summary). MR 570590
B. J. Gardner, Simple rings whose lower radicals are atoms, Acta Math. Hungar. 43 (1984), no. 1-2, 131–135. MR 731971, DOI 10.1007/BF01951333
W. G. Leavitt and L. C. A. van Leeuwen, Rings isomorphic with all proper factor-rings, Ring theory (Proc. Antwerp Conf. (NATO Adv. Study Inst.), Univ. Antwerp, Antwerp, 1978) Lecture Notes in Pure and Appl. Math., vol. 51, Dekker, New York, 1979, pp. 783–798. MR 563323
E. R. Puczyłowski, Some questions concerning radicals of associative rings, Theory of radicals (Szekszárd, 1991) Colloq. Math. Soc. János Bolyai, vol. 61, North-Holland, Amsterdam, 1993, pp. 209–227. MR 1243913
Edmund R. Puczyłowski and Ewa Roszkowska, Atoms of lattices of radicals of associative rings, Radical theory (Sendai, 1988) Uchida Rokakuho, Tokyo, 1989, pp. 123–134. MR 999585
Edmund R. Puczyłowski and Ewa Roszkowska, On atoms and coatoms of lattices of radicals of associative rings, Comm. Algebra 20 (1992), no. 4, 955–977. MR 1154396, DOI 10.1080/00927879208824385
Robert L. Snider, Lattices of radicals, Pacific J. Math. 40 (1972), 207–220. MR 308175, DOI 10.2140/pjm.1972.40.207
J. F. Watters, Non-commutative minimally non-Noetherian rings, Math. Scand. 40 (1977), no. 2, 176–182. MR 457489, DOI 10.7146/math.scand.a-11686