Ok. Have a question:

Which of the following six sets are subrings of M(R)? Which ones have an identity?

M(R)=matrix(a b

c d)

(a) All matrices of the form

(0 r

0 0) with r in Q

(b)

(a b

0 c) with a,b,c in Z

(c)

(a a

b b) a, b, in R

(d)

(a 0

a 0) a in R

(e)

(a 0

0 a) a in R

(f)

(a 0

0 0) a in R

Ok. I think I need to prove just that assuming a,b are in the subset that a+b, ab, and -a are also in the subset. Also that 0,1 are in the subset.

I did a few, but they take a while. I figured that (a) is a subring without identity and (b) is a subring with identity. The book says that (c) is a subring with identity, but what is the identity matrix? Can anyone confirm these answers and possible ones for (d) (e) and (f)? (e) appears to be a subring with identity.