In his book Module Theory (1977), Blyth defines the column rank of an $m\times n$ matrix $A$ over a commutative unitary ring $R$ to be
the dimension of the subspace of $\mathrm{Mat}_{m\times 1}(R)$ generated by the column...matrices of $A$
and dually for row rank (p. 153). However, he only defines dimension for free modules over commutative unitary rings (p. 105), so I don't think this definition will work as stated in general (even after changing "subspace" to "submodule") because a submodule of even a finitely generated free module over a commutative unitary ring need not be free.
I'm trying to determine how to interpret an exercise which asks to show that the matrix $$A=\begin{bmatrix}1&2&3\\0&3&2\end{bmatrix}$$ over the ring $\mathbb{Z}/30\mathbb{Z}$ has a row rank of 2 and a column rank of 1 (p. 170). Should I just interpret this to be referring to the maximal numbers of linearly independent rows/columns?
Note in the latest edition of his book (2018), it seems he has stated the definitions of rank for matrices over arbitrary unitary rings $R$ (p. 111), and generalized the definition of dimension to include also free modules over division rings (p. 78). The same exercise still appears (p. 121).