  We consider two natural models of random walks on a module $V$ over a finite
commutative ring $R$ driven simultaneously by addition of random elements in
$V$, and multiplication by random elements in $R$. In the coin-toss walk,
either one of the two operations is performed depending on the flip of a coin.
In the affine walk, random elements $a \in R,b \in V$ are sampled
independently, and the current state $x$ is taken to $ax+b$. For both models,
we obtain the complete spectrum of the transition matrix from the
representation theory of the monoid of all affine maps on $V$ under a suitable
hypothesis on the measure on $V$ (the measure on $R$ can be arbitrary).
