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 spectrum, the Pruefer property, separation of points and polynomial density. We also study the stable rank (a fundamental concept in algebraic K-theory) of rings of integer-valued polynomials. 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).

In this project we conducted fundamental research in abstract algebra, such as group theory and ring theory, and used group theory and ring theory to answer questions about factorization in rings of integer-valued polynomials and questions about the structure of groups of polynomial permutations on finite rings.