Is testing whether the nonunits of $K[x,y]_{(x,y)}[1/y]$ form an ideal a messy undertaking or am I going about it wrong?

Question

I'm studying the ideals of $R = K[x,y]_{(x,y)}[1/y]$: the polynomials in $1/y$ with coefficients in the localization of $K[x,y]$ at the ideal $(x,y)$. I've managed to identify that its units are given by members $h_1/h_2 \in R$ who have their numerator $h_1$ in $\langle y \rangle (K[x,y] - {(x,y)}) $, understood as the set of nonnegative powers of $y$ multiplied by an element in $K[x,y]$ that is not in $(x,y)$, so now I'm trying to figure out if the nonunits, given by $$I_? = \{h_1/h_2 \in K[x,y]_{(x,y)}[1/y]: h_1 \notin \langle y \rangle (K[x,y] -{(x,y)}) \}$$ form an ideal, and I'm working on a proof that involves proving closure under substraction, since nonunits are multiplicatively closed, but laboriously expanding and carefully arguing about the form of the resulting polynomials is starting to get messy. Is there a better way to go about this?