Suppose that $R^n \cong M \oplus N$ where $R$ is a commutative ring and $M$ and $N$ are $R$-modules.
Consider a map of the form $f \oplus 0_N: M \oplus N \to M \oplus N$ where $f: M \to M$ is a linear map.
This can be represented by some $n\times n$ matrix over $R$, but do we know anything about how certain parts of this matrix look?
That is, is there some basis for $M\oplus N$ for which the matrix representing $f\oplus 0_N$ has some known form?
If we're in the category of vector spaces, then it is obvious, but in our case $M$ and $N$ may not be free.
I know this question is a bit vague, but any insight is much appreciated.