I've been asked to show that -
If $A = \mathbb{R}[x,y,z]/(x^2+y^2+z^2-1)$ then $z-1$ is a prime in $A$. (Here $z-1$ means the corresponding coset of $z-1$ in $A$.)
For this I quotiented by the ideal $(z-1)$ which gives $\mathbb{R}[x,y]/(x^2+y^2)$ How do I show that this is an integral domain? (It'll imply $(z-1)$ is a prime ideal and hence $z-1$ is a prime.)
So you've reduced your problem to showing that $x^2 + y^2$ is prime in $\mathbb{R}[x,y]$.
To make things even simpler, remember that in $\mathbb{R}[x,y]$, being prime and being irreducible are the same thing (follows, for example, from $\mathbb{R}[x,y]$ being a UFD).
Can you show that no non-constant polynomials divide $x^2 + y^2$?