TITLE: Non-isomorphic Heisenberg groups over rings
QUESTION [10 upvotes]: Suppose $R_1,R_2$ are finite unital commutative rings. Consider Heisenberg groups $H_3(R_1)$ and $H_3(R_2)$ (upper unitriangular marticies $3 \times 3$). 

Proposition. If $R_1 \not\cong R_2$ (as rings) then $H_3(R_1) \not\cong H_3(R_2)$ (as multiplicative groups). 

Is that true? It seem that if $R_1$ and $R_2$ have not isomorphic additive groups then $H_3(R_1) \not\cong H_3(R_2)$, since they have not isomorphic centers. But what about general case, or is that too broad?

REPLY [11 votes]: 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 $K^{\mathrm{op}}$.