## 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].

### MSC:

 13G05 Integral domains 13B25 Polynomials over commutative rings

### Keywords:

point separation; interpolation domain