  Let $\mathfrak D$ be a residually finite Dedekind domain, $a\in \mathfrak D$
be a nonzero element and $\mathfrak n$ be a nonzero ideal of $\mathfrak D$. In
this paper we describe the dynamics of the map $x\mapsto ax$ over the quotient
ring $\mathfrak D/\mathfrak n$. We further present some applications of our
main result.
