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Residually small varieties ofK-algebras

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Abstract

We focus on those universal algebras which ring theorists call associative algebras overK, whereK is a commutative ring having an identity element. They are calledK-algebras in this paper. For fixedK there are two full varieties to consider namelyK-algebras with, or without, an identity element as a formal constant.

A varietyV is calledresidually small if there is a cardinal λ such thatV=SP(V λ), that is, every algebra inV is isomorphic to a subalgebra of a product of algebras which have cardinality at most λ.

The result of this paper characterizes residually small varieties ofK-algebras in a way that is independent ofK. Namely,V (any variety ofK-algebras with or without identity element) is residually small if and only if one of the identities (x−xn)(y−yn)=[(x−xn)(y−yn)]n is a law inV (for somen>1). Our proofs combine universal algebraic ideas with standard techniques of ring theory and yield a complete description of the subdirectly irreducible algebras in these varieties.

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Authors and Affiliations

  1. University of California, Berkeley, California, USA

    Ralph McKenzie

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  1. Ralph McKenzie
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Research Supported by U.S. National Science Foundation grant number MCS 77-22913.

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McKenzie, R. Residually small varieties ofK-algebras. Algebra Universalis 14, 181–196 (1982). https://doi.org/10.1007/BF02483919

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  • DOI: https://doi.org/10.1007/BF02483919

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