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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.

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    Regarding localization in the non-commutative setting, you might want to consider [this MO answer](http://mathoverflow.net/questions/7917/non-commutative-algebraic-geometry/15196#15196) which gives some intuition as to why localization can be problematic in the non-commutative setting. Regards,2012-08-02
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    Thanks for the link. I still think that my example is one of the easiest case and we can say something by localization.2012-08-02

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