I have a system of homogeneous linear equations over $\mathbb{Z}_6$. The coefficients of the variables are all 1 (maybe this helps). The number of equations is very small compared to the number of variables (more precisely, I can keep the number of equations constant while the number of variables can be as high as I wish).
I want to say that there are "a lot" of solutions to this equation system, and over a field it would have been easy; the rank of the corresponding matrix would have been small compared to the number of variables, and so the dimension of the solution space would have been high.
However, over $\mathbb{Z}_6$ all the "traditional" linear algebra I know ceases to work. So, can I still use similar claims? If so, how? Is there a book dealing with linear algebra over rings (maybe not general rings; $\mathbb{Z_n}$ is all I need).