$\mathbb{R}[x,y,z]/(x^2+y^2+z^2)$ is a field or just an integral domain

Question

In Steps in Commutative Algebra of Sharp, Problem 3.24 implies $\mathbb{R}[x,y,z]/(x^2+y^2+z^2)$ is an integral domain. But is it a field? Since the ideal is irreducible, I can't find a bigger ideal.

Answer 1

The ideal $I=(x^2+y^2+z^2)$ is homogeneous so the ring $\mathbb R[x,y,z]/I$ is a graded ring, with degrees in $\mathbb N_0$. In particular, only its elements of degree $0$ —which are the scalars— can be invertible. Since the ring is not just $\mathbb R$ (for example, its homogeneous component of degre $1$ is $3$-dimensional) it is not a field.

Answer 2

You should take $(x^2,y^2,z^2)$ which is larger than $(x^2+y^2+z^2)$.