Root closure
type: Item
instance of: scholarly article
title: Root closure
zbMATH Open document ID: 0541.13002
DOI: 10.1016/0021-8693(84)90205-9
author: Gerhard Angermüller
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 14M05
zbMATH DE Number: 3859263
zbMATH Keywords: root closure
zbMATH Keywords: formal power series
zbMATH Keywords: normality of variety
zbMATH Keywords: ring extensions
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90205-9
OpenAlex ID: W4210599358
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description: scientific article; zbMATH DE number 3859263