Finite normalizing extensions
Let the ring \(S\) be a finite normalizing extension of the ring \(R\). The author establishes a number of basic facts for this set-up, among which are descent of flatness (that is, if \((E\otimes_ R S)_ S\) is flat, then \(E_ R\) is flat) and that \(S\otimes_ RE=0\) implies that \(E=0\). The latter provides a very simple proof of properness, that is, \(SN\) is a proper left ideal of \(S\) whenever \(N\) is a proper left ideal of \(R\). The descent of projectivity is also established, and for the case when \(S_ R\) is flat, generalizations of these descents to homological dimensions are obtained. In particular, it is shown that if the left global dimension of \(R\) is finite, then it is bounded from above by that of \(S\).
scientific article; zbMATH DE number 130657
scientific article