Finite simple abelian algebras are strictly simple
MATTHEW A. VALERIOTE · Proceedings of the American Mathematical Society · 1990
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.