Showing that an ideal is maximal Let $k$ be an algebraically closed field and $f$ be the polynomial $x_1x_2+x_2x_3+x_3x_1$ in $k[x_1, x_2, x_3]$. Here $f$ is irreducible.
Then this polynomial ring is not a $PID$, it is only an $UFD$. But I have to show that the ideal generated by $f$ is maximal in $k[x_1, x_2, x_3]$. Could anyone help me with this?

The ideal generated by $f$ is not maximal.  It is properly contained in the maximal ideal $(x_1,x_2,x_3)$.
