Question

Prove that in every commutative ring of characterstic 2 (a+b)^2=a^2+b^2

Answer 1

A proof
0=(a+a)^2-(a+a)=(a^2-a)+(a^2-a)+a+a=a+a=0
As to your point regarding commutatively note that R must be commutative

(a+b)^2=a^2+ab+ba+b^2
=(a+b)
ab+ba=0