Geodesic
On the coincidence of the factor and Gondran--Minoux rank functions of matrices over a~semiring
Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 6, pp. 223-232.

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We consider the rank functions of matrices over semiring, functions that generalize the classical notion of the rank of a matrix over a field. We study semirings over which the factor and Gondran–Minoux ranks of any matrix coincide. It is shown that every semiring satisfying that condition is a subsemiring of a field. We provide an example of an integral domain over which the factor and Gondran–Minoux ranks are different. 
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Ya. N. Shitov. On the coincidence of the factor and Gondran--Minoux rank functions of matrices over a~semiring. Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 6, pp. 223-232. http://geodesic.mathdoc.fr/item/FPM_2012_17_6_a6/

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