Normal elements and completions of non-commutative Noetherian rings.

*(English)*Zbl 0627.16013Let R be a Noetherian ring with a prime ideal P and let R’ denote the P- adic completion of R. If R is sufficiently non-commutative, then P and R’ are necessarily badly behaved compared with what happens in the commutative case. More precisely, suppose from now on that there is a regular normal element a of R such that \(aPa^{-1}\neq P\). Then P does not satisfy the Artin-Rees property. Also classical localization at P is not possible, and a belongs to all the symbolic powers of P. An example shows that even in these circumstances R’ can be Noetherian. Suppose further that the intersection of the powers of P is 0, that a is regular as an element of R’, and that \((aPa^{-1}+P)/P\) is not contained in the Jacobson radical of R/P. Then R’ is not Noetherian and P-adic completion is not an exact functor on finitely-generated left R-modules.

Reviewer: A.W.Chatters

##### MSC:

16P40 | Noetherian rings and modules (associative rings and algebras) |

16P50 | Localization and associative Noetherian rings |

16Dxx | Modules, bimodules and ideals in associative algebras |