Dimension of Factor Ring

**hurle32** · March 16th 2011, 03:59 PM

Find the dimension of $\mathbb{R}[x,y]/(2xy,x+y+1).$

**NonCommAlg** · March 16th 2011, 10:58 PM

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.

**hurle32** · March 16th 2011, 11:16 PM

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!

**NonCommAlg** · March 16th 2011, 11:35 PM

$y+I=-x-1 + I$ 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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