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HNN-extensions of algebras and applications

Published online by Cambridge University Press:  17 April 2009

Hans-Christian Mez
Hans-Christian Mez
Affiliation:
Department of Mathematics, University of California, Irvine, California 92717, USA.
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The classic HNN-embedding theorem for groups does not transfer to associative rings or algebras. In its first part this paper presents constructions which provide such a theorem if an additional condition is put on the isomorphic subalgebras or if one restricts to algebras over fields and drops the associativity. The main part of the paper deals with applications of these results. For example, it is known that every existentially closed group is ω-homogeneous. It is shown that the corresponding is false for existentially closed associative Δ-algebras but true for existentially universal nonassociative K-algebras. Further-more, orthogonal sequences of idempotents in existentially closed associative Δ-algebras over a regular ring Δ are investigated. It is shown that the conjugacy class of such a sequence depends only on a corresponding order sequence. In particular, in every existentially closed K-algebra all idempotents different from 0 and 1 are conjugated.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 29 , Issue 2 , April 1984 , pp. 215 - 229
Copyright
Copyright © Australian Mathematical Society 1984

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