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Ring monomorphism from M(2;R)-M(3;R)

  1. Feb 4, 2010 #1
    1. The problem statement, all variables and given/known data
    Give an example of a ring monomorphism f:M(2;R)-M(3;R)


    2. Relevant equations



    3. The attempt at a solution
    I can't think of anything that would be a monomorphism.
     
    Last edited: Feb 4, 2010
  2. jcsd
  3. Feb 5, 2010 #2
    I assume you mean the rings [tex]M_2(\mathbb{R})[/tex] and [tex]M_3(\mathbb{R})[/tex] of respectively [tex]2 \times 2[/tex] and [tex]3 \times 3[/tex] matrices with real entries.

    Think of a monomorphism as an "embedding", i.e., how can you embed the [tex]2\times 2[/tex] matrices in the [tex]3\times 3[/tex] ones without changing their structure?

    One way to attack this is to use geometry. A [tex]2\times 2[/tex] matrix [tex]A[/tex] is a linear transformation of [tex]\mathbb{R}^2[/tex]. How can you extend [tex]A[/tex] to be a linear transformation of [tex]\mathbb{R}^3[/tex], in such a way that you don't "interfere with" the action of [tex]A[/tex] on [tex]\mathbb{R}^2[/tex] in any way?
     
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