Let's take an $R$-module $M$ and two finite presentations $$R^n\xrightarrow{f} R^m\to M\to 0$$and$$R^a\xrightarrow{g} R^b\to M\to 0$$
The two morphisms $f$ and $g$ are equivalently two matrix that we call matrix presentations of $M$. I wish to know if it exists some relation between $f$ and $g$.
I'm also interested in the particular case in which $a=n$ and $b=m$, if it makes any difference.
We have Schanuel's lemma if $K_f$ and $K_g$ are the kernels of $f$ and $g$, i.e. if we have the exact sequences: $$0\longrightarrow K_f\longrightarrow R^n\xrightarrow{\,\;f\;\,} R^m\longrightarrow M\longrightarrow 0$$ $$0\longrightarrow K_g\longrightarrow R^a\xrightarrow{\,\;g\;\,} R^b\longrightarrow M\longrightarrow 0,$$ then, there exists an isomorphism $$K_f\oplus R^a\oplus R^m\simeq K_g\oplus R^n\oplus R^b.$$