Showing two given rings are isomorphic

Question

Let $K$ be a field and $R=K[x,y]$ be a polynomial ring in the variable $x$ and $y$. let $R_1=R[y/x]$ is a subring of the quotient field of $R$. Let $R_2=R[t]/(xt-y)$. show that $R_1$ and $R_2$ are isomorphic. The problem that I can show that there exists a homomorphism from $R[t]$ to $R_1$ by sending $t$ to $y/x$. But I am unable to show that the kernel of this homorphism is indeed equal to $(xt-y)$. Any kind of help will be appreciated. Thanks in advance.

Answer 1

Let us denote your map by $\phi$. We can prove this by showing two inclusions: $(xt-y) \subseteq \text{Ker}(\phi)$ and $\text{Ker}(\phi) \subseteq (xt-y)$. I think the first one you can look at yourself.

For the second one, take an arbitrary $f \in R[t]$ such that $f \in \text{Ker}(\phi)$.
Then $f(y/x) = 0$. This means that $f(t) = 0 \bmod (xt-y)$, therefore $f(t)$ can be factored as $(xt-y) \cdot g(t)$, for some $g \in R[t]$. Hence $f \in (xt-y)$.