Semisimplicity of the algebra of indecomposable representations of a finite commutative semigroup

Finite semigroups S, having only a finite number of indecomposable matrix representations over a field K of characteristic are considered. It is proved that the algebra over the field F of K-representations of the semigroup S is semisimple if and only if for any maximal subgroup of the semigroup S the characteristic of the field F does not divide the index of its Sylow p-subgroup and is greater than the order of this Sylow subgroup.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 57, pp. 125–128, 1976.