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

  1. Jun 20, 2011 #1
    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.

  2. jcsd
  3. Jun 20, 2011 #2
    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...
  4. Jun 20, 2011 #3
    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.
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