Based on arXiv:math/9802041v1arXiv:math/9802041v1, there is a definition for $NC$-filtration and $NC$-completion of an associated algebra over the complex numbers:
Let $R$ be an associative algebra and $R^{\rm Lie} = (R, [a,b] = ab - ba)$ be $R$ regarded as a ${\rm Lie}$ algebra. By Definitiondefinition the following hold:
(a) The NC-filtration of $R$ is the decreasing filtration $\{F^dR\}_{d\ge 0}$ where $F^dR$ is the two-sided ideal $$F^dR = \sum_m \sum_{i_1+\cdots+ i_m -m = d} R \cdot R^{\rm Lie}_{i_1} \cdot R \cdot{\kern 1.5pt}{\cdots}{\kern 1.5pt}\cdot R\cdot R^{\rm Lie}_{i_m}\cdot R$$$$F^dR = \sum_m \sum_{i_1+\cdots+ i_m -m = d} R \cdot R^{\rm Lie}_{i_1} \cdot R \cdot{\kern 1.5pt}{\cdots}{\kern 1.5pt}\cdot R\cdot R^{\rm Lie}_{i_m}\cdot R.$$
(b) For an associative algebra $R$ its NC-completion is the algebra $$R_{[ ab ]} = \lim_{\leftarrow} R/F^dR$$$$R_{[ ab ]} = \lim_{\leftarrow} R/F^dR.$$ My question is: How can we generalize this definition to a Noetherian $R$-algebra $A$, when $R$ is a commutative noetherianNoetherian ring.?
