Dimension of an integer valued polynomial ring (Q923633)

Let \(A\) be a commutative domain with quotient field \(K\). Define \(A_s=\{f\in K[X] \mid f(A)\subseteq A\}\), the ring of ``integer valued'' polynomials. The author shows that the Krull dimension of \(A_s\) satisfies the relation \(\dim (A_s)\ge \dim (A[X])-1\). He gives some examples where equality holds, and remarks that it is not known if strict inequality holds (non-trivially). He also gives a criterion for \(A_s\) to be different to \(A[X]\).