Let $k$ denote a field.
If $\varphi$ is a $k$-linear map between $k$-modules freely generated by finite sets, then the kernel of $\varphi$ can easily be found by reinterpreting $\varphi$ as a matrix and performing row-reduction.
Suppose instead that $\varphi$ is a $k$-algebra homomorphism between $k$-algebras freely generated by finite sets, i.e. between polynomial algebras in finitely-many indeterminates. Is there a way of explicitly computing the kernel of such a map?
Let's look at a really simple example. Suppose we want to show that $$\frac{\mathbb{R}[x,y]}{(y-x^2)} \cong \mathbb{R}[x],$$
for example. A reasonable place to start is to define $\varphi$ to be the unique morphism of $\mathbb{R}$-algebras $$\mathbb{R}[x,y] \rightarrow \mathbb{R}[x]$$ satisfying $\varphi(x)=x$ and $\varphi(y) = x^2$. This is clearly surjective, and presumably the kernel of this map is $(y-x^2),$ thereby giving the desired result via the first isomorphism theorem.
Part of my question is: if $\varphi$ were more complicated, how could we work out what it's kernel was? Is there a systematic approach, like there is in linear algebra?
Part of my question is also about rigor. In the particular example of $\varphi$ given above, since $\varphi(x) = x$, thus $\varphi(x^2) = x^2$, so $\varphi(y - x^2) = 0$. Thus $(y-x^2) \subseteq \mathrm{ker}(\varphi)$. I'm a bit unsure how to go the other way. Supposing $\varphi(P) = 0$, how can we show that $P \in (y - x^2)$? And is there a method that works generally, even for much more complicate examples?