Properties of Gl(n,R); R a ring/division ring

[Bacle]

Hi, All:

Could someone please tell me or give me a ref. on the basic properties of

Gl(n,R) ; R a ring; possibly a division ring, and Gl(n,R) the group (under composition)

of matrices invertible over R ? (I imagine we need a ring R with 1 , to talk about

invertibility). I mostly would like to see how the properties of Gl(n,R) are different

from those of Gl(n,F) , where F is a field.

Thanks.

 

[henry_m]

For a commutative ring, a matrix is invertible if and only if its determinant is a unit.
If that ring is a field, every nonzero element is a unit so we recover the well-known result that a matrix is invertible if and only if it has nonvanishing determinant.
Example: a matrix over the integers has an inverse over the integers if and only if its determinant is plus or minus 1.

As for noncommutative rings, I really don't know anything...

 

[Bacle]

Wow, I'm keeping you busy today, henry_m.

Do you know anything about orthogonal and symplectic groups

associated to Gl(n,R)? I mean, we have an R-module R_M , and

symplectic /quadratic forms q_S , q_Q respectively . Then the symplectic/orthogonal group

associated with (R_M,Q) is defined to be the subgroup of Gl(n,R) that preserves q_S, resp. q_Q.