When Does (ab)^n Equal (a^n)(b^n) in Ring Theory?

[gunnygunz]

Homework Statement

Let R be a ring and a,b be elements of R. Let m and n be positive integers. Under what conditions is it true that (ab)^n = (a^n)(b^n)?

Homework Equations

The Attempt at a Solution

We must show ab = ba.

Suppose n = 2.

Then (ab)^2 = (ab)(ab) = a(ba)b = a(ab)b = (aa)(bb) = (a^2)(b^2).

I am not sure where to go from here...

 

[jbunniii]

First of all, you said "let m and n be integers," but m was never mentioned again. Is there a typo somewhere?

Second: You said "We must show ab = ba". But then you ASSUMED it was true in this step: a(ba)b = a(ab)b. This is not necessarily true unless the ring is commutative, in which case this problem becomes trivial.

I would start with the n = 2 case. What must be true in order for (ab)^2 to equal a^2 b^2? This is equivalent to writing

abab = aabb

or

a(ba - ab)b = 0

Clearly this is true if ab = ba, but is this necessary? What if this is a ring of matrices, for example?