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.