Are Ideals of Mn(Z) Commutative?

[DukeSteve]

Hello Experts,

Again a Q and what I did, please tell me what I am doing wrong:

Given that there is a ring of matrices above Z (integers) Mn(Z) and 2 ideals I, J of this ring.

I need to prove that they are commutative: IJ = JI

What I did is that:

For all i in I and for all M in Mn(Z) n is the the size of a matrix n x n

M*i in I and i*M is also in I.

same with J : j*M in J and M*j is in J

For every k in J and for every h in I:

kh = j*M*i*M = j*(M*i)*M = ... I don't know what to do from here... please guide me.

 

[gilsonrfilho]

Hey Duke!
I suggest you to check again your hypothesys instead.
Well, \mathbb{Z} is a ring with unity, right? What's the form of it's ideals? What's the form of the ideals of M_{n}(\mathbb{Z})?

The problem of your approach is that I can't really see a way to use your hypothesys.