Invertible matrices and semilinear spaces over commutative semirings

This paper deals with invertible matrices and semilinear spaces over commutative semirings. Some necessary and sufficient conditions for matrices to be invertible over commutative semirings are obtained. The condition under which the cardinality of each basis of semilinear space of n-dimensional vectors is n is given. In the end, the factor rank of matrices over semirings is investigated.