Subrings invariant under endomorphisms. (Q1868915)

Let \(S\) and \(R\) be rings of regular functions of affine algebraic varieties over a field of characteristic \(0\), and \(R\) be a subring of \(S\). Suppose that every ideal of height \(1\) in \(R\) generates a proper ideal in \(S\), and Spec\,\(R\) has no self-intersection points. Let \(F:S\rightarrow S\) be an endomorphism such that \(F(R)\subset R\). The author shows that if \(F\) is an automorphism, then \(F| _{R}:R\rightarrow R\) is an automorphism as well. It is also shown that if \(R\) and \(S\) have the same transcendence degree, then the fact that \(F| _{R}\) is an automorphism implies that \(F\) is an automorphism.