I'm trying to learn by myself some algebraic geometry, since I will probably be taking an elementary course next semester.
I have some trouble understanding the following:
Let $Y=Z(y-x^{2})$ then by definition $Y$ is algebraic set in $\mathbb{A}^{2}$. The author finds the coordinate ring $K[x,y]/I(Y)$ where $I(Y)=\{f \in K[x,y]: f=0 \ \textrm{in Y}\}$.
Claim: $K[x,y]/I(y) \cong K[t]$ where $t$ is an indeterminate.
Define a map $\phi: K[x,y] \rightarrow K[t]$ by $x \mapsto t$ and $y \mapsto t^{2}$. Then $f$ is a ring morphism by the universal property.
Questions:
1) Why it suffices to say where to send $x$ and $y$?
2) How do you check using the universal property that this is indeed a ring morphism? I don't know how to apply this property
3) What is the reason of sending $x$ to $t$ and $y \mapsto t^{2}$,i.e how do you figure out the images of $x$ and $y$ under $\phi$ ?
I would really appreciate if you can please explain as much as possible, hope that is not too much to ask.
Thanks in advance