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It's a theorem of Garland that $K_2(R)$ is finite. Perhaps the best way to get a handle on it is to use Quillen's localization sequence $$0\rightarrow K_2(R)\rightarrow K_2(F)\stackrel{T}{\rightarrow} \oplus_v k(v)^*\rightarrow 0,$$ where $F$ is the fraction field and the $k(v)$ are the residue fields. The map $T$ is the sum of the tame symbols, which is surjective by a theorem of Matsumoto. The injectivity on the left follows from the vanishing of $K_2$ for finite fields. This isn't much of an answer, but considering $K_2(R)$ as a subgroup of $K_2(F)$ seems a reasonable way to start some concrete considerations.