  Let $\mathbb{Z}_{p^s}$ be the residue class ring of integers modulo $p^s$,
where $p$ is a prime number and $s$ is a positive integer. Using matrix
representation and the inner rank of a matrix, we study the intersection, join,
dimension formula and dual subspaces on vector subspaces of
$\mathbb{Z}^n_{p^s}$. Based on these results, we investigate the Grassmann
graph $G_{p^s}(n,m)$ over $\mathbb{Z}_{p^s}$. $G_{p^s}(n,m)$ is a connected
vertex-transitive graph, and we determine its valency, clique number and
maximum cliques. Finally, we characterize the automorphisms of $G_{p^s}(n,m)$.
