Finite simple abelian algebras are strictly simple

Valeriote, Matthew A.

doi:10.1090/S0002-9939-1990-0990434-2

Finite simple abelian algebras are strictly simple
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by Matthew A. Valeriote PDF

Proc. Amer. Math. Soc. 108 (1990), 49-57 Request permission

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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