Geodesic
Finiteness conditions for subdirectly irreducible $S$-acts and modules
Fundamentalʹnaâ i prikladnaâ matematika, Tome 4 (1998) no. 2, pp. 763-767.

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It is proved that, for every semigroup $S$ of $n$ elements, the cardinalities of the subdirectly irreducible $S$-acts are less or equal to $2^{n+1}$. If the cardinalities of the subdirectly irreducible $S$-acts are bounded by a natural number then $S$ is a periodic semigroup. It is obtained a combinatorial proof of the fact that there exist only finitely many of unitary subdirect irreducible modules over a finite ring. 
@article{FPM_1998_4_2_a21,
     author = {I. B. Kozhukhov},
     title = {Finiteness conditions for subdirectly irreducible $S$-acts and modules},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {763--767},
     publisher = {mathdoc},
     volume = {4},
     number = {2},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1998_4_2_a21/}
}
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I. B. Kozhukhov. Finiteness conditions for subdirectly irreducible $S$-acts and modules. Fundamentalʹnaâ i prikladnaâ matematika, Tome 4 (1998) no. 2, pp. 763-767. http://geodesic.mathdoc.fr/item/FPM_1998_4_2_a21/