I was asked the following question during my oral exam (seminar course on commutative algebra), but I couldn't answer. I would like to learn how to solve such problems.
Prove/disprove that $I = \langle X^2+Y^2-Z^2+XY\rangle$ is a prime ideal in $R = k[X,Y,Z]$, where $k$ is a field. Also, prove that the Krull dimension of the quotient ring $R/I$ is $1$.
This course has been very advance for me (most of Atiyah-MacDonald was covered), and I am a slow learner. Hence a detailed discussion about such problems will be appreciated.
Edit: based on the comments of MooS and Bernard it's clear that I remembered the second part of the problem wrongly. It should be:
Prove that the Krull dimension of the quotient ring $R/I$ is $2$, or equivalently that $\:\operatorname{Proj}R/I$ has dimension 1.