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Distributive factor lattices in free rings


Author: P. M. Cohn
Journal: Proc. Amer. Math. Soc. 105 (1989), 34-41
MSC: Primary 16A06
DOI: https://doi.org/10.1090/S0002-9939-1989-0973837-3
MathSciNet review: 973837
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Abstract: For any field $ E$ with subfield $ k$ the free $ E$-ring over $ k$ on a set $ X,\quad R = {\text{ }}{E_k}\left\langle X \right\rangle $ is a fir. It is proved here that when $ E/k$ is purely inseparable, then the submodule lattice $ R/cR$ is distributive, for any $ c \ne 0$ ( $ R$ has distributive factor lattice); by contrast this is false when $ E/k$ is a nontrivial Galois extension and $ X \ne \emptyset $.


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

  • [1] G. M. Bergman, Commuting elements in free algebras and related topics in ring theory, Thesis, Harvard Univ., 1967.
  • [2] -, Modules over coproducts of rings, Trans. Amer. Math. Soc. 200 (1974), 1-32. MR 0357502 (50:9970)
  • [3] P. M. Cohn, Algebra 2, Wiley, Chichester, 1977, Second edition in preparation.
  • [4] -, Ringe mit distributivem Faktorverband, Abh. Braunschweig. Wiss. Ges. 33 (1982), 35-40. MR 693161 (85a:16004)
  • [5] -, Free rings and their relations, 2nd ed., London Math Soc. Monographs, no. 19, Academic Press, London, Orlando, 1985. MR 800091 (87e:16006)
  • [6] W. Dicks and E. D. Sontag, Sylvester domains, J. Pure Appl. Algebra 13 (1978), 243-275. MR 509164 (80j:16014)
  • [7] M. Eichler, Einführung in die Theorie der algebraischen Zahlen und Funktionen, Birkhäuser-Verlag, Basel, 1963. MR 0168561 (29:5821)

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DOI: https://doi.org/10.1090/S0002-9939-1989-0973837-3
Article copyright: © Copyright 1989 American Mathematical Society

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