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].
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Notes
Communicated by I. Shestakov.