Q:

To Prove That Field of Fractions of Given Rings is Same. Proof Verification.

I am trying to solve Q. 8a in Section 9.1 from Abstract Algebra by Dummit & Foote.
The problem is:

Let $F$ be a field and $R=F[x,x^2y,x^3y^2,...,x^ny^{n-1},...]$ be a subring of $F[x,y]$. Show that the field of fractions of $R$ and $F[x,y]$ are the same.

My Proof:
Say the field of fractions of $R$ is $K$.
Since $R$ is contained in $F[x,y]$, $K$ is contained int he field of fractions of $F[x,y]$.
But since $x,x^2y\in K$, we have $x^2y/x^2=y\in K$.
Thus $x,y\in K$.
So each element of the field of fractions of $F[x,y]$ is in $K$, showing the reverse containment.

I am apprehensive about my solution because by my reasoning, the field of fractions of $F[x,x^2y]$ is same as the field of fractions of $F[x,y]$.
Further, the solution given here is much more complicated.
Can anybody confirm that my proof is correct or else find a flaw in it?

A:

Your proof is fine - if you wish to be more precise, you may show that the smallest subfield of $k := \text{Frac}(F[x,y])$ containing both $x, y$ is $k$. When writing statements such as $x^2y/x^2 = y$, keep in mind that everything is viewed as a subfield of $k$, and it is inside $k$ that elements are being compared (e.g. there is an equality of fractions $\dfrac{x^2y}{x^2} = \dfrac{y}{1}$).
You are also correct that $\text{Frac}(F[x,x^2y]) = k$, by the same reasoning.

