Isomorphic general linear groups not implies isomorphic rings

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Statement

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:

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

Related facts

References