# Thread: COOL Abstract Algebra Homework!!! :D

1. ## COOL Abstract Algebra Homework!!! :D

#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

2. Originally Posted by GOKILL
#1
Let R be a ring where $\displaystyle \forall a,b,c\in\mathbb{R}\text{ satisfying }a^2+b^2+c^2=ab+bc+ac$.
If (i) R is commutative and (ii) 1+1 and 1+1+1 have inverse in R (1 is unity in R), show that 1R1=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
both your question have problems!! you should check your question more carefully before submitting it:

in question 1, 1R1 = 1 has no meaning. besides if in $\displaystyle a^2+b^2+c^2=ab+ac+bc$ you put a = 2 = 1+1 and b = c = 1 you'll get 1 = 0 and so R ={0}!!

in question 2, did you mean $\displaystyle \mathbb{I}=\langle x^2-1 \rangle$?

3. I know nothing about Ring Theory but i'd still like to say something here .

$\displaystyle a^2 + b^2 + c^2 = ab + bc +ca$
$\displaystyle 2a^2 + 2b^2 + 2c^2 = 2ab + 2bc +2ca$
$\displaystyle (a-b)^2 + (b-c)^2 + (c-a)^2 = 0 \implies a=b=c$