Noetherian completion and PI rings.
The author studies a question raised by A. Goldie. Let \(R\) be a Noetherian ring with a two-sided ideal \(I\). When is the \(I\)-adic completion of \(R\), \(\widehat R=\varprojlim_i R/I^i\), Noetherian? Partial answers, applications, examples given by various authors for special families of rings and ideals are presented in [\textit{A. Braun}, Proc. Am. Math. Soc. 138, No. 3, 847-852 (2010; Zbl 1218.16010)]. The main result of the paper is the following characterization. Theorem. Let \(R\) be a left Noetherian PI ring and \(I\) a two-sided ideal in \(R\) which is right finitely generated and such that \(R/I\) is right Noetherian. Let \[ \mathcal F=\{P\in\roman{spec\,}R\mid\bigcap_i(I^i+P)=P,\;\mathrm{r-ann}_R[(I\cap P)/\bigcap_i(IP+I^i)]=P\}. \] Then the following statements are equivalent: 1) \(\widehat R\) is left Noetherian; 2) For each \(P\in\mathcal F\) either \(I\subseteq P\) or \(R/P\) is complete in its induced \(I\)-adic topology.
scientific article; zbMATH DE number 6509335
scientific article