  We consider irreversible Markov chains on finite commutative rings randomly
generated using both addition and multiplication. We restrict ourselves to the
case where the addition is uniformly random and multiplication is arbitrary.
  We first prove formulas for eigenvalues and multiplicities of the transition
matrices of these chains using the character theory of finite abelian groups.
The examples of principal ideal rings (such as $\mathbb{Z}_{n}$) and finite
chain rings (such as $\mathbb{Z}_{p^k}$) are particularly illuminating and are
treated separately. We then prove a recursive formula for the stationary
probabilities for any ring, and use it to prove explicit formulas for the
probabilities for finite chain rings when multiplication is also uniformly
random. Finally, we prove constant mixing time for our chains using coupling.
