Let $M$ be a finitely generated module over a commutative ring $R$. The first order deformation of module $M$ is parametrized by $Ext^{1}(M,M)$ and the obstruction is parametrized by $Ext^{2}(M,M)$. Is there a similar story for noncommutative $R$? I don't expect this to be true for any noncommutative rings but wonder if this is still true for some $good$ ones. I would appreciate any reference suggestion, comments, and ideas.

Edit I would like naively to compute the tangent space of the moduli space $X$ of module with some data if it exists. At $M\in X$, the tangent space can be understood as the set of extensions of the $R$-module $M$ to some $R\otimes_{k} k[\epsilon]/(\epsilon^2)$-module. The obstruction is defined in the same manner.

2 typos corrected

Let $M$ be a finitely generated module over a commutative ring $R$. The first order deformation of module $M$ is parametrized by $Ext^{1}(E,E)$ Ext^{1}(M,M)$and the obstruction is parametrized by$Ext^{2}(E,E)$. Ext^{2}(M,M)$. Is there a similar story for noncommutative $R$? I don't expect this to be true for any noncommutative rings but wonder if this is still true for some $good$ ones. I would appreciate any reference suggestion, comments, and ideas.

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# Deformation of modules over noncommutaitve rings

Let $M$ be a finitely generated module over a commutative ring $R$. The first order deformation of module $M$ is parametrized by $Ext^{1}(E,E)$ and the obstruction is parametrized by $Ext^{2}(E,E)$. Is there a similar story for noncommutative $R$? I don't expect this to be true for any noncommutative rings but wonder if this is still true for some $good$ ones. I would appreciate any reference suggestion, comments, and ideas.