I am trying to convince myself of an isomorphism between:
$$k[x,y,z]/(x^2-yz,z-1) \rightarrow k[t]$$
In trying to show that these rings are isomorphic, I have constructed a map sending: $x \rightarrow t, y \rightarrow t^2, z \rightarrow 1$. Now this map is clearly surjective, and I'm pretty certain that the kernel of this map is indeed (the ideal): $(x^2-yz,z-1)$, however I'm not entirely certain how I would prove that it is...Is this the best way to show that these rings are isomorphic?
Any help would be greatly appreciated!