Question

Let R be a ring:

Prove that (a^2-b^2) = (a-b)(a+b) iff R is a commmutative ring

Answer 1

one way should be easy
if R is commutative then
\((a-b)(a+b)=a^2+ab-ba-b^2=a^2+ab-ab+b^2 \text{ because commutative }\)
\(=a^2-b^2\)

Answer 2

the other way should not be hard either
if \((a+b)(a-b)=a^2-b^2\) then \(ab-ba=0\) and so \(ab=ba\) as needed

Answer 3

Thanks:)