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Math Help - COOL Abstract Algebra Homework!!! :D

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

    #1
    Let R be a ring where \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
    \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.
    \mathbb{I}=<x^2+1> True/False? Explain it!

    Thanks for your help
    Last edited by GOKILL; May 20th 2010 at 12:47 AM. Reason: |R|=1 means the cardinality of R is 1
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  2. #2
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    Quote Originally Posted by GOKILL View Post
    #1
    Let R be a ring where \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
    \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.
    \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 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 \mathbb{I}=\langle x^2-1 \rangle?
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  3. #3
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    I know nothing about Ring Theory but i'd still like to say something here .

    Can this help you ?

     a^2 + b^2 + c^2 = ab + bc +ca

     2a^2 + 2b^2 + 2c^2 = 2ab + 2bc +2ca

     (a-b)^2 + (b-c)^2 + (c-a)^2 = 0 \implies a=b=c
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  4. #4
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    You should put on clarity the effort you put to posting copies of the same question in two different forums (here) and suffering lack of clarity on both.
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