On ranks of matrices over noncommutative domains

We consider matrices with entries in some domain $R$, i.e., in a ring, not necessarily commutative, not containing non-trivial zero divisors. The concepts of the row rank and the column rank are discussed. (Coefficients of linear dependencies belong to the domain $R$; left coefficients are used for rows, right coefficients for columns.) Assuming that the domain satisfies the Ore conditions, i.e., the existence of non-zero left and right common multiples for arbitrary non-zero elements, it is proven that these row and column ranks are equal, which allows us to speak about the rank of a matrix without specifying which rank (row or column) is meant. In fact, the existence of non-zero left and right common multiples for arbitrary non-zero elements of $R$ is a necessary and sufficient condition for the equality of the row and column ranks of an arbitrary matrix over $R$. An algorithm for calculating the rank of a given matrix is proposed. Our Maple implementation of this algorithm covers the domains of differential and ($q$-)difference operators, both ordinary and with partial derivatives and differences.