  We extend our previous study of Markov chains on finite commutative rings
(arXiv:1605.05089) to arbitrary finite rings with identity. At each step, we
either add or multiply by a randomly chosen element of the ring, where the
addition (resp. multiplication) distribution is uniform (resp. conjugacy
invariant). We prove explicit formulas for some of the eigenvalues of the
transition matrix and give lower bounds on their multiplicities. We also give
recursive formulas for the stationary distribution and prove that the mixing
time is bounded by an absolute constant. For the matrix rings $M_2(\mathbb
F_q),$ we compute the entire spectrum explicitly using the representation
theory of $\text{GL}_2(\mathbb F_q),$ as well as the stationary probabilities.
