FWF - Poly-Alg - Integer-valued polynomials on algebras

We investigate two kinds of rings of polynomials mapping elements of an algebra to elements of the same algebra. Both generalize the classical concept of integer-valued polynomials. An example of the first, commutative, kind of ring is the ring of polynomials with rational coefficients mapping each n by n integer matrix to an integer matrix. An example of the second, non-commutative, kind of ring is the ring of polynomials with coefficients in the algebra of n by n rational matrices, mapping every n by n integer matrix to an integer matrix. As with the classical integer-valued polynomials, the main questions concern integral-closure, the Pruefer property, separation of points, polynomial density and non-unique factorization. Both kinds of integer-valued polynomials on algebras require innovative combinations of techniques from commutative ring theory and non-commutative ring theory for their study. Also, they are useful for constructing rings with certain specified properties (for instance, with respect to factorization).