Artin-Rees Lemma for Non-Commutative Rings Explained

Question

Consider a commutative noetherian ring $A$ with an ideal $I\subset A$. The Artin-Rees lemma implies that for f.g. modules $N\subset M$, the $I$-adic topology on $N$ agrees with the subspace topology coming from the $I$-adic topology on $M$. I wonder how much this can be generalized to the case where $A$ is not necessarily commutative. For example, let $\mathcal{R}$ be the valuation ring of a complete, discretely valued non-archimedean field $K$, and let $\pi$ be a uniformizer of $\mathcal{R}$. Assume you have a non-commutative algebra $A$ over $\mathcal{R}$ which is two-sided noetherian, and complete with respect to the $\pi$-adic topology. Are there any conditions on $A$ which assure that the Artin-Rees lemma holds for the $\pi$-adic topology? Notice that the previous statement covers, for example,
the case where $A$ is the $\pi$-adic completion of the universal enveloping algebra of a finitely generated Lie algebra over $\mathcal{R}$.

Answer 1

There is some discussion of this in Rowen's "Ring Theory", volume I, Section 3.5, with additional references therein.

Exercise 19 on p. 462 in op. cit. states that a polycentral ideal $I$ of a noetherian ring $A$ has the Artin-Rees property, i.e., for every f.g. left module $M$ and a f.g. submodule $N\subseteq M$, there is $i\geq 1$ with $N\cap I^i M\subseteq IN$. (Applying this to $I^nN$ in place of $N$ shows that the $I$-adic topology on $N$ conicides with the topology induced from the $I$-adic topology of $M$.)

Here, an ideal $I$ is called polycentral if there is a chain of ideals $I=I_0\supseteq I_1\supseteq\dots\supseteq I_{t+1}=0$ such that $I_i = I_{i+1} +\sum_{r=1}^{s(i)} a_r A$ with $a_1,\dots a_{s(i)}\in A$ central modulo $I_{i+1}$. In particular, any ideal generated by central elements is polycentral.

In the context of your question, $I=\pi A$ is polycentral since it is generated by a central element, so it has the Artin-Rees property.

Actually, it is noted in op. cit. that the usual proof of the Artin-Rees lemma in the commutative case carries over in the non-commutative case for ideals generated by central elements.