# Commutative Ring

• February 21st 2011, 03:16 PM
DanielThrice
Commutative Ring
My professor gave us this query at the end of class, and I started thinking about axioms, but I didn't think of anything that would have to make the following statement true. Thoughts?

What must be true about a nontrivial commutative ring R in order to conclude
(a+b)^4 = a^4 + 2a^2b^2 + b^4 for all elements a and b in R?

Clarification would be great on this one.
• February 21st 2011, 05:43 PM
roninpro
Note that by the binomial theorem,

$(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$

If the ring has characteristic 4, that is, $4x=0$ for all $x\in R$, then this would simplify to

$a^4+2a^2b^2+b^4$
• February 22nd 2011, 11:38 AM
DanielThrice
So that just means it has to be characteristic 4?