Theorem 1.13 of this paper (The model theory of unitriangular groups, Ann. Pure Appl. Logic 68(3), 1994, 225-261) by O. Belegradek says that the answer is positive.
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 $K^{\mathrm{op}}$.
