Noetherian completion and PI rings

Let R be a Noetherian PI ring with a two-sided ideal I. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat R$$\end{document} be the I-adic completion of R. We establish a criterion which determines when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat R$$\end{document} is Noetherian, thus answering a question of A. Goldie. As a consequence a question of J. McConnell is also answered.