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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Finite simple abelian algebras are strictly simple


Author: Matthew A. Valeriote
Journal: Proc. Amer. Math. Soc. 108 (1990), 49-57
MSC: Primary 08A30; Secondary 03C05
DOI: https://doi.org/10.1090/S0002-9939-1990-0990434-2
MathSciNet review: 990434
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Abstract: A finite universal algebra is called strictly simple if it is simple and has no nontrivial subalgebras. An algebra is said to be Abelian if for every term $ t(x,\bar y)$ and for all elements $ a,b,\bar c,\bar d$, we have the following implication: $ t(a,\bar c) = t(a,\bar d) \to t(b,\bar c) = t(b,\bar d)$. It is shown that every finite simple Abelian universal algebra is strictly simple. This generalizes a well-known fact about Abelian groups and modules.


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DOI: https://doi.org/10.1090/S0002-9939-1990-0990434-2
Article copyright: © Copyright 1990 American Mathematical Society

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