Diagonalization over $\mathbb{Z}$

Question

I'm studying knot theory and the textbook mentions that to find the mod $p$ rank of a knot, we take a certain matrix $M$ with integer entries corresponding to the knot and diagonalize it over $\mathbb{Z}$, then count the number of entries on the resulting matrix that are $0$ mod $p$. I'm not too familiar with linear algebra -- what does it mean to diagonalize a matrix over $\mathbb{Z}$ and how is this distinct for normal diagonalization (over $\mathbb{R}$)?

Answer 1

Its distinct. The point is that the ring of integers has just to units $\pm1$ for which inverses exist; on the other hand, e.g., $2$ is not invertible as $1/2$ is not an integers. This hampers or makes diagonalization eventually impossible.

In a field like the field of real numbers, all nonzero elements are invertible making diagonalization possible.