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matrix ring

i need to show that a matrix ring (nxn matricies with coeffecients in ring R) is commutative if and only if R is a commutative ring and n=1.

I know how to start with the end and show the first, but i'm stuck with the other way. I know matrix multiplication is non-commutative so only 1x1 matricies will commute and then once you know n=1 then i think you can deduce that R is isomorphic to the matrix ring hence is commutative. But i'm not sure how to write this algebraically, or even if it is right?

Thanks! :-)
Original post by Ishika
i need to show that a matrix ring (nxn matricies with coeffecients in ring R) is commutative if and only if R is a commutative ring and n=1.

I know how to start with the end and show the first, but i'm stuck with the other way. I know matrix multiplication is non-commutative so only 1x1 matricies will commute and then once you know n=1 then i think you can deduce that R is isomorphic to the matrix ring hence is commutative. But i'm not sure how to write this algebraically, or even if it is right?

Thanks! :-)


What if there exist a ring R such that for some n, all matrices with elements in R commute? You need to show that there does not.

Spoiler

Reply 2
Avatar for Ishika
Ishika
OP
I think i know how to show that R is commutative but could you give me a hint on how to show that n=1. In the question it says assume the ring is non-trivial so i think i need to assume that n doesn't equal 1 and then arrive at R={0} to get a contradiction - could you give me a hint?
Reply 3
Avatar for DFranklin
DFranklin
18
If your definition of ring includes means R must contain 1 (i.e. multiplicative identity), then:

If n > 1, define E_ij to be the matrix with 1 in the E_ij position and 0 elsewhere. If you choose two suitable matrices E_ij E_kl, you'll find EijEkl≠EklEijE_{ij} E_{kl} \neq E_{kl} E_{ij}.

(It's probably easiest to look at the 2x2 case - once you've found the matrices for that case it's easy to generalise).

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