Closed Polynomials in Polynomial Rings over Unique Factorization Domains

Let R be a UFD, and let M(R, n) be the set of all subalgebras of the form R[f], where f ∈ R[x 1 ,..., x n ]∖R. For a polynomial f ∈ R[x 1 ,..., x n ]∖R, we prove that R[f] is a maximal element of M(R, n) if and only if it is integrally closed in R[x 1 ,..., x n ] and Q(R)[f] ∩ R[x 1 ,..., x n ] = R[...

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Bibliographic Details
Published inCommunications in algebra Vol. 43; no. 5; pp. 1935 - 1938
Main Authors Kato, Masaya, Kojima, Hideo
Format Journal Article
LanguageEnglish
Published Taylor & Francis Group 04.05.2015
Subjects
Closed polynomial
Derivation
Primary: 13B25
Secondary: 13N15
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Summary:Let R be a UFD, and let M(R, n) be the set of all subalgebras of the form R[f], where f ∈ R[x 1 ,..., x n ]∖R. For a polynomial f ∈ R[x 1 ,..., x n ]∖R, we prove that R[f] is a maximal element of M(R, n) if and only if it is integrally closed in R[x 1 ,..., x n ] and Q(R)[f] ∩ R[x 1 ,..., x n ] = R[f]. Moreover, we prove that, in the case where the characteristic of R equals zero, R[f] is a maximal element of M(R, n) if and only if there exists an R-derivation on R[x 1 ,..., x n ] whose kernel equals R[f].
ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2013.879876