Let $R=\mathbb{C}[x,y,z]/(x^3+y^3+z^3)$ be the coordinate ring of the affine variety defined by the equation $x^3+y^3+z^3=0$. We can consider the localization in the element $x$, denoted by $R_x$. I have to compute $Spec(R_x)$. We have surely maximal ideals that we can find by choosing $(a,b,c) \in \mathbb{C}^3$ such that $a^3+b^3+c^3=0$ and $a \ne 0$. Then we can find 12 lines: $$ (t,0,bt), \,\,\,\, (bt,0,t), \,\,\,\, (t,bt,0), \,\,\,\, (bt,t,0) $$ where $t \ne 0$ and $b^3=-1$. So we have the correspondent ideals $$ I_1=(y,bz-x), \,\,\,\, I_2=(y,z-bx), \,\,\,\, I_3=(z,by-x), \,\,\,\, I_4=(z,y-bx) .$$ Now I have two questions:
1) Are there other lines? I'm looking for irreducible varieties of $R_x$, so I have to prove that $R_x/I_i$ is a domain for $i=1,\dots, 4$. How can I do it?
2) If I see $R_x \subset \mathbb{P}^2_{\mathbb{C}}$ I think that $Spec(R_x)=\{(0), points\}$, where points (maximal ideals) correspond to the lines in the affine space $\mathbb{A}^3_{\mathbb{C}}$. Therefore is it true that the 12 lines that I found are the intersection points of the cubic with the projective axis?