Alizter
commented
4 months ago @ThomatoTomato Would you like to have a go at defining the general linear group once #2005 is merged? It should be in theories/Algebra/Groups/GL.v and depend on Algebra.Rings.Ring.
2005 adds notions of invertibility for rings. It also defines the group of units of a ring. This should allow us to define $\mathrm{GL}_n(R)$ over an arbitrary ring.
I'm not sure what else we can prove about $\mathrm{GL}_n(R)$ for an arbitrary ring $R$, once we have determinants it would be possible to define $\mathrm{SL}_n(R)$ and show we have an exact sequence of groups:
$$ 1 \to \mathrm{SL}_n(R) \to \mathrm{GL}_n(R) \to R^\times \to 1 $$