Let $R$ be the ring $\mathbb{Z}[X,Y,Z]/(Y − X + 1, Y − Z + 2, 3X^2 - YZ + 3X +2Y + 4)$. What are the solutions $e \in R$ of the equation $e^2 = e$ with $e \not\in \{0,1\}$?
So $e$ has to be a residue class of a polynomial $f \in \mathbb{Z}[X,Y,Z]$ modulo the ideal $(Y − X + 1, Y − Z + 2, 3X^2 - Y Z + 3X + 2Y +4)$. Could someone explain to me how to find at least one solution?
I have already shown that $R$ is isomorphic to $\mathbb{Z} \times \mathbb{Z}[T]/(2T+3)$, but I'm not sure you need it to solve my question?