Let \(k\) be an infinite field with a nonzero element \(q\) and \(\mathcal O_q(\mathrm{Mat}(m\times n,k))\) the coordinate algebra of quantum \(m\times n\)-matrices with entries from \(k\) and with one parameter \(q\). The first section of the paper presents a survey of results on ring-theoretical properties of \(\mathcal O_q(\mathrm{Mat}(m\times n,k))\). These rings are usually defined in terms of generators and defining relations. The author proposes another approach using the deleting derivations algorithm due to G. Cauchon. This approach is applied to the classification of torus-invariant prime ideals in \(\mathcal O_q(\mathrm{Mat}(m\times n,k))\). It is proved that if \(q\) is not a root of 1 any such ideal is generated by quantum minors. Moreover it is shown that quantum minors form a Gröbner basis of a torus-invariant prime ideal.