Completions of Noetherian P.I. rings
Let R be a left noetherian p.i. ring, and let M be a semimaximal ideal of R, the intersection of whose powers is zero. The author proves that \(\hat R=\lim_{\leftarrow i}R/M^ i\), the completion of R with respect to \(\{M^ i\}\), is left noetherian, thus answering a question that goes back to \textit{A. W. Goldie} [J. Algebra 5, 89-105 (1967; Zbl 0154.288)]. He also shows that the height of M may be different from the height of \(\hat M=\hat RM\hat R\). Furthermore, \(\hat R\) is not necessarily flat over R as it is in the commutative case. In fact, for a maximal ideal M, it is shown that \(\hat R\) is flat as a left R-module precisely when M is right localizable.
scientific article; zbMATH DE number 4154666
scientific article