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 $3 \times 3$). 



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> **Proposition.** If $R\_1 \not\cong R\_2$ (as rings) then $H\_3(R\_1) \not\cong H\_3(R\_2)$ (as multiplicative groups). 
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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?



Non-isomorphic Heisenberg groups over rings
Theorem 1.13 of [this paper](http://www.sciencedirect.com/science/article/pii/0168007294900221) *(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 $K^{\mathrm{op}}$.


