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Zbl 0675.13006
Cahen, Paul Jean
Anneaux presque intégralement clos. (Almost integrally closed rings). (French)
[J] Ann. Sci. Univ. Blaise Pascal Clermont-Ferrand II 91, Math. 24, 61-64 (1987). ISSN 0249-7042

A commutative integral domain A with quotient field K is said to be {\it almost integrally closed} if given any $x\in K$ such that $x\sp n\in A$ for some positive integer n then $x\in A$. Clearly any integrally closed domain is almost integrally closed by an exercise of {\it N. Bourbaki} gives an example to show that the converse is false [Éléments de mathématique. Fasc. XXX: ``Algèbre commutative''. Chapitre 5: ``Entiers'' (1964; Zbl 0205.343); $exercise\quad 15,$ p. 75)]. \par The author provides further examples by considering quadratic extensions of the field of rational numbers and, generalising the Bourbaki example, the subring B of the polynomial ring L[x] consisting of those polynomials with constant term in K, where L/K is a field extension.
[J.Clark]

MSC 2000:: *13B20
13B02 Extension theory (commutative rings)

Keywords: almost integrally closed rings; quadratic extensions of the field of rational numbers; polynomial ring

Citations: Zbl 0547.13002; Zbl 0205.343

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