The proof that column rank = row rank for matrices over a field relies on the fact that the elements of a field commutate. I'm looking for an easy example of a matrix over a ring for which column rank $\neq$ row rank. E.g. can one find a $2 \times 3$-(block)matrix with real $2\times 2$-matrices as elements, which has different column and row ranks?

The proof that column rank = row rank for matrices over a field relies on the fact that the elements of a field commute. I'm looking for an easy example of a matrix over a ring for which column rank $\neq$ row rank. i.e. can one find a $2 \times 3$-(block)matrix with real $2\times 2$-matrices as elements, which has different column and row ranks?