Isomorphic general linear groups not implies isomorphic rings

Statement for $$n = 2$$
It is possible to have two commutative unital rings $$R_1,R_2$$ such that the general linear group of degree two $$GL(2,R_1)$$ is isomorphic to the general linear group of degree two $$GL(2,R_2)$$, but $$R_1$$ and $$R_2$$ are not isomorphic as rings.

Two notable examples are:


 * $$GL(2,\mathbb{Z}/4\mathbb{Z}) \cong GL(2,\mathbb{F}_2[t]/(t^2))$$: See general linear group:GL(2,Z4).
 * $$GL(2,\mathbb{Z}/9\mathbb{Z}) \cong GL(2,\mathbb{F}_3[t]/(t^2))$$: See general linear group:GL(2,Z9).

(These are the only examples for finite discrete valuation rings, it seems).

Related facts

 * Isomorphic general linear groups implies isomorphic fields