Is $C[x,y,z]/<y-x^2,z-x^3>$ an Integral Domain?

  • #1
evinda
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Hi! (Smile)

Can I justify like that, that $C[x,y,z]/<y-x^2,z-x^3>$ is an integral domain?

We show that $ker(\phi)=<y-x^2,z-x^3>$.

From the theorem of isomorphism, we have that $C[x,y,z]/<y-x^2,z-x^3> \cong im(\phi)$

$im(\phi)$ is a subring of $\mathbb{C}[x]$

$\mathbb{C}[x]$ is an integral domain, since $\mathbb{C}$ is a field.

As $im(\phi)$ is a subring of $\mathbb{C}[x]$, $im(\phi)$ is also an integral domain.

(Thinking)
 
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  • #2
Hi evinda,

You need to define some morphism to claim such things.
 

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