Answer to Question #267640 in Abstract Algebra for Aman

Question

Prove for any ring R and a,b∈ R , (a+b)²= a²+2ab+b²

Answer 1

Any commutative ring $R$ and $a,b\in R$ , $(a+b)^2=a^2+2ab+b^2$

Prove:

$(a+b)^2=(a+b)(a+b)=a(a+b)+b(a+b)=\\ =a\cdot a+a\cdot b+b\cdot a+b\cdot b=\\ =a^2+a\cdot b+b\cdot a+b^2\\ (a+b)^2=(a+b)(a+b)\\ a^2+2ab+b^2=a^2+ab+ba+b^2\\ ab=ba$

That a ring $R$ is commutative