Theorem 1.13 of [this paper][1] <i>(The model theory of unitriangular groups, Ann. Pure Appl. Logic
68(3), 1994, 225-261)</i> by O. Belegradek says that the answer is positive even for infinite commutative rings. 


However, Proposition 1.9 in the same paper asserts that this does not extend to the non-commutative (associative unital case). The counterexamples have the form $R_1=K\times K$, $R_2=K\times K^{\mathrm{op}}$, where $K$ is indecomposable and not isomorphic to $K^{\mathrm{op}}$.

  [1]: http://www.sciencedirect.com/science/article/pii/0168007294900221