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Finitely generated simple algebras: A question of B. I. Plotkin

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Abstract

In his recent series of lectures, Prof. B. I. Plotkin discussed geometrical properties of the variety of associativeK-algebras. In particular, he studied geometrically noetherian and logically noetherian algebras and, in this connection, he asked whether there exist uncountably many simpleK-algebras with a fixed finite number of generators. We answer this question in the affirmative using both crossed product constructions and HNN extensions of division rings. Specifically, we show that there exist uncountably many nonisomorphic 4-generator simple Ore domains, and also uncountably many nonisomorphic division algebras having 2 generators as a division algebra.

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin-Parkside, 53141, Kenosha, WI, USA

    A. I. Lichtman

  2. Department of Mathematics, University of Wisconsin-Madison, 53706, Madison, WI, USA

    D. S. Passman

Authors
  1. A. I. Lichtman
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  2. D. S. Passman
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Additional information

The first author is grateful to Professor B. I. Plotkin for communicating this problem to him and for stimulating conversations.

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Lichtman, A.I., Passman, D.S. Finitely generated simple algebras: A question of B. I. Plotkin. Isr. J. Math. 143, 341–359 (2004). https://doi.org/10.1007/BF02803506

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  • DOI: https://doi.org/10.1007/BF02803506

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