Let $k$ be a field. Find the dimension of $k[x,y,z]/(xyz,y^2)$.
Let $R=k[x,y,z]$ and $I=(xyz,y^2)$ then a prime ideal in $R/I$ is in correspondence with a prime ideal $\mathfrak{p}$ in $R$ containing $I$. So, $y\in\mathfrak{p}$. Then $\mathfrak{p}$ corresponds to a prime ideal of $K[x,z]\{=(k[x,y,z]/(xyz,y^2))/(y)\}$. Hence the dimension of $R/I$ is $2$. Is this a correct argument?
If not, then could someone point out the mistake and kindly show me the correct way to approach this problem?