Suppose a hypersurface in $\mathbb{P}^n$ is given by an equation $F(Z) = 0$. It is easy to show that polynomials $Z_i F$ ($i = 0,\ldots,n$) give the same hypersurface, but I have trouble demonstrating that $K[Z_0,\ldots,Z_n]/I \cong K[Z_0, \ldots, Z_n]/I'$ where $I = (F)$, $I' = (Z_i F \mid i = 0,\ldots,n)$.
I tried to construct an isomorphism explicitly, but I got stuck: intuitively, $\varphi: G \mapsto \sum_{i=0}^n Z_i G$ seems like it could be right, it is defined correctly because if $G = FH$, then $\varphi(G) = \sum_i Z_i F H \in I'$.
But then I have to show that $\varphi$ is surjective, and I'm stuck: I can't find a way to show that for any polynomial $G$ we have $G \equiv \sum_i Z_i F \: (\mathrm{mod}\ I')$.
Am I on the right track? Is there an easier way to show that $K[Z]/I \cong K[Z]/I'$?