Thread: 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.

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.