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What does it mean for a Ring to be Stabilized by a map

  1. Oct 8, 2012 #1
    1. The problem statement, all variables and given/known data
    Let D be a division ring, C its center and let S be a division subring of D which is stabilized by every map x -> dxd-1, d≠0 in D. Show that either S = D or S is a subset of C.


    2. The attempt at a solution
    I havent actually started working on it yet because I am not sure what it asking. What does it mean for a subring to be stabilized by a map? And what is x? Is it an element of D?

    I am familiar with the stabilizer of an element, but the above terminology is confusing me.
     
  2. jcsd
  3. Oct 9, 2012 #2

    quasar987

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    It means that dSd-1=S for all nonzero d. (same definition as the stabilizer of an element but with the set S substituted in its place!)

    I think a more common terminology would be to say

    "...and let S be a division subring of D which is stable under the action by every map x -> dxd-1, d≠0 in D"
     
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