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 $K^{\mathrm{op}}$.

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 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 $K^{\mathrm{op}}$.

The proof of Proposition 1.9 of this paper http://ac.els-cdn.com/0168007294900221/1-s2.0-0168007294900221-main.pdf?_tid=5d45c892-b816-11e6-8401-00000aab0f27&acdnat=1480631867_bccc7147520bd9d7356358bb364ec8e1 by Belgradek says the answer is no.

However Theorem 1.13 says the answer is yes if the rings are commutative.

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 $K^{\mathrm{op}}$.