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Math Help - Dimension of Factor Ring

  1. #1
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    Dimension of Factor Ring

    Find the dimension of \mathbb{R}[x,y]/(2xy,x+y+1).
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  2. #2
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    if by "dimension" you mean dimension as a vector space over \mathbb{R}, the answer is 2. this is easy to see: put x+y+1=u and 2xy=v and let I be the ideal of \mathbb{R}[x,y] which is generated by u and v. show that x^2+x \in I and conclude that \{1+I, x+I\} is an \mathbb{R}-basis for your ring.
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  3. #3
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    I get everything up to the last conclusion: why does x^2 + x \in I imply that  \{1+I,x+I\} is a basis?

    Thanks!
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  4. #4
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    y+I=-x-1 + I and so every element of your ring is in the form p(x)+I, for some p(x) \in \mathbb{R}[x]. also, since x^2 + x \in I, we have p(x)+I=ax + b + I, for some a,b \in \mathbb{R}. so \{1+I, x + I\} generates your ring. it is obvious that the set is \mathbb{R}-linearly independent.
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