A b s t r a c t.
Let $\cal V$ be a variety of algebras with a finite list of finite directly
indecomposable members. We show that there is a polynomial time algorithm
that tests the isomorphism between any two finite algebras from $\cal V.$
This includes the following classical structures in algebra:
Abelian groups with $nx=0$, $n>0$,
Rings with $x^m=x$, $m>1$,
Modules over a finite semisimple ring.
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