Polynomial separation of points in algebras. (English) Zbl 1092.13027

Chapman, Scott T. (ed.), Arithmetical properties of commutative rings and monoids. Boca Raton, FL: Chapman & Hall/CRC (ISBN 0-8247-2327-9/pbk). Lecture Notes in Pure and Applied Mathematics 241, 253-259 (2005).
Let \(D\) be an integral domain with field of quotients \(K\), let \(\text{Int}(D)\) be the ring of all polynomials \(f\in K[X]\) mapping \(D\) into \(D\), and let \(M_2(D)\) be the ring of \(2\times2\) matrices with entries in \(D\). It is shown that \(D\) is an interpolation domain if and only if whenever \(a,b\) are algebraic elements of a \(K\)-algebra \(A\) with co-prime minimal polynomials in \(K[X]\), then there exists a polynomial \(f\in \text{Int}(D)\) with \(f(a)=0\) and \(f(b)=1\). It is also proved that one cannot replace here the condition \(f\in \text{Int}(D)\) by \(f\in \text{M\,int}_2(D)\), \(\text{M\,int}_2(D)\) being the set of polynomials \(g\in K[X]\) with \(g(M_2(D))\subset M_2(D)\).
For the entire collection see [Zbl 1061.13001].


13G05 Integral domains
13B25 Polynomials over commutative rings