Root closure
For an integer \(n>1\), a subring R of a ring S is said to be n-root closed in S if whenever \(x\in S\) with \(x^ n\in R\), then \(x\in R\); reference to S is omitted, if S is the total quotient ring of R. Generalizing earlier results [cf. the author, J. Algebra 83, 437-441 (1983; Zbl 0516.13006)], it is shown that a variety over an algebraically closed field of characteristic \(p\geq 0\) with at most finitely many non-normal points is normal iff its local rings are n-root closed for some \(n>1\), \(n\not\equiv 0(mod p).\)- In the second part of the paper it is proven that the property of being n-root closed for some \(n>1\) (resp. integrally closed) is stable under passage to power series for ring extensions \(R\subseteq S,\) where R is an integral domain with quotient field K such that \(K\cap S=R\) [for extensions \(R\subseteq S\) with R absolutely flat, cf. \textit{J. J. Watkins}, J. Algebra 75, 43-58 (1982; Zbl 0531.13003)]. - The paper is closed with the observation that for reduced rings \(R\subseteq S\) the property of being n-root closed for some \(n>1\) (resp. seminormal) is stable under passage to \(R(X)\subseteq S(X),\) the localizations of the polynomial rings with respect to the set of all polynomials, whose coefficients generate the unity ideal [for an analogous result concerning integrally closedness, cf. \textit{T. Akiba}, J. Math. Kyoto Univ. 20, 749- 752 (1980; Zbl 0463.13003)].
scientific article; zbMATH DE number 3859263