I'm trying to show that the following ring is not local: $R = \frac {k[x,y,z]}{\left<(x-1)z\right>}$ where $k$ is an algebraically closed field. I guess I'm searching for some ideals that will result in a field but I can't see how to do this.
[EDIT: Thanks to MooS for the correction]
Take the quotient of $R$ by the ideal $(x-z)$, and that ideal contains $(x-1)(x-z)+(x-1)z=(x-1)x$, and you can show $x$ is a nontrivial idempotent of the quotient.
So, $R/(x-z)$ has a nontrivial idempotent $\implies$ $R/(x-z)$ is not local $\implies$ $R$ is not local.