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Math Help - Field

  1. #1
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    Nov 2006
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    Field

    Yo. This problem is really bothering me. I could use some help.
    Let F be a field and let R, the ring be the following matrix
    a11=a11
    a12=a12
    a13=a13
    a21=0
    a22=a22
    a23=a23
    a31=0
    a32=0
    a33=a33
    where aij is in F, the field.
    Let I={(aij) in R : a11=a22=a33=0}.
    Prove that R is a subring of the 3 by 3 matrix M3(F). Also show that I is an ideal of R and R/I=F x F x F.
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  2. #2
    Global Moderator

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    I have been thinking about this problem.
    If I get bored I might start working on it but the idea is this.

    First, R is a ring. That is simple to show just show that all the ring defintions are satisfied. Lengthy, but easy.

    Second, to show that I is an ideal of R is also not no hard. Show that I is an additive subgroup. And then show that aI and Ia are subsets of I. However, if you are smart you can skip this step and go to step 3 which will answer this question.

    Third, is the tricky part. To show the Isomorphism between R/I and F x F x F. What you need is to define a ring homomorphism from R to F x F x F in such a way that the kernel is I and that phi [R]=F x F x F. Then by the fundamental homomorphism theorem we have not only shown that I is an ideal but settled that R/I is isomorphic with F x F x F.
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