Closed Polynomials in Polynomial Rings over Unique Factorization Domains

Abstract

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].

Key Words:

• Closed polynomial
• Derivation

2010 Mathematics Subject Classification:

• Primary: 13B25
• Secondary: 13N15

Notes

Communicated by I. Shestakov.