Thread: Subring

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.

It appears very clear, I think, that the identity referred to here should be the multiplicative identity. Every ring has an additive identity. In the case of these matrices if the following matrix is present then it has the multiplicative identity
[1 0]
[0 1].
Thus it clear that (a), (c), (d), & (f) cannot have that element.
On the other hand, both (b) & (e) can have that element.

But you say that the book says (c) has identity. Well if that is true the book may be talking about the additive identity! In that case each of the sets contains the element
[0 0]
[0 0].

Thanks for the help. Maybe the book just has a typo for c and that is what's throwing me off.

Also, are all of these subrings? They appear to be (to me at least).

Originally Posted by JaysFan31

Also, are all of these subrings? They appear to be (to me at least).

It appears to me too. I did not formally do each one out cuz it takes too long, but experience tells me that they should be. The only thing I each wether each one is closed under multiplication.