I am looking for references for solving systems of linear equations $Ax = b \bmod m$, i.e., $Ax=b$ where all entries are in the ring ${\mathbb Z}/m{\mathbb Z}$.
I know that when $m$ is prime then we have a field and so standard linear algebra applies. I know also that we can use the Chinese remainder theorem to reduce to the case where $m$ is a prime power. That's where I am stuck.
I am hoping there is some canonical textbook (or perhaps a chapter in an algebra or linear algebra textbook) that covers this material comprehensively, e.g., defining appropriate notions of linear independence, rank, bases, etc. that are applicable to this case.
I know also that this problem is connected more generally to module theory and/or general linear equations over commutative rings. If you are aware of an introductory-level treatment of that material that would help here, that is also ok. But I would actually prefer not to get bogged down in a general treatment if I can avoid it, because all I care about is the modular case.