Let $A$ be the non-commutative ring given by $ A=\mathbb{C}\langle x,y,z \rangle /(xy=ayx,yz=bzy,zx=cxz) $ for some $a,b,c\in \mathbb{C}$. What is the localization $A_{(x)}$ of A with respect to the (two-sided) ideal $(x)$? If it can be defined, is it graded ring? In general what condition is required to localize a ring?
I think of $A$ as a non-commutative $\mathbb{P}^2$ and wonder whether or not we can study it by local patch.
Thank you in advance.
I should mention this; my main problem is I don't know about the definition of localization. Moreover even if it is defined I am not sure whether this technique is useful or not. I would like to conclude for example smoothness of the non-commutative $\mathbb{P}^2$ or a hypersurface in it. I hope to confirm this by checking it locally.