Geodesic
On the property of being Frobenius of the semigroup algebra of a~finite commutative semigroup
Izvestiya. Mathematics , Tome 2 (1968) no. 4, pp. 781-797.

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In this article we give a criterion for the semigroup algebra $K\mathfrak G$ of a finite commutative semigroup $\mathfrak G$ over a field $K$ to be Frobenius. 
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     title = {On the property of being {Frobenius} of the semigroup algebra of a~finite commutative semigroup},
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I. S. Ponizovskii. On the property of being Frobenius of the semigroup algebra of a~finite commutative semigroup. Izvestiya. Mathematics , Tome 2 (1968) no. 4, pp. 781-797. http://geodesic.mathdoc.fr/item/IM2_1968_2_4_a6/

[1] Curtis C. W., Reiner T., Representation theory of finite groups and associative algebras, New York, London, 1962 | MR

[2] Lyapin E. S., Polugruppy, Fizmatgiz, M., 1960 | MR