What is equivalence?

Two matrices $B, C \in M_{n,k}(A)$ are equivalent if $C = QBP$ with $Q \in GL_n(A)$, $P \in GL_k(A)$ (invertible over $A$: determinant in $A^\times$). Equivalent presentation matrices define isomorphic modules $A^n/ \operatorname{im}$ (change bases in $A^n$ and $A^k$).