#1

Let R be a ring where $\displaystyle \forall a,b,c\in R\text{ satisfying }a^2+b^2+c^2=ab+bc+ac$.

If (i) R is commutative and (ii) 1+1 and 1+1+1 have inverse in R (1 is unity in R), show that |R|=1.

#2

$\displaystyle \mathbb{I}\subseteq\mathbb{R}[x]\text{, where }\mathbb{I}$ is the set of all polynomials whose the sum of even degree coefficients is 0 and the sum of odd degree coefficients is 0.

$\displaystyle \mathbb{I}=<x^2+1>$ True/False? Explain it!

Thanks for your help