Polynomial functions in commutative ring theory

Our project concerns commutative ring theory, with connections to topology, arithmetic, group theory and algebraic K-theory. Concerning topological methods in commutative ring theory, we plan to characterize non- Archimedean uniformities by a list of equivalent axioms, and, with the added insight, to investigate completions of rings with respect to I-adic topologies, and their applications to Skolem properties and the prime spectrum of rings of functions. Concerning the arithmetic of Prüfer and Krull rings, we will study questions on non-unique factorization of elements into irreducibles in non-Noetherian Prüfer rings; for instance, the existence or non-existence of prime elements and absolutely irreducible elements, and also, divisor theories and divisor homomorphisms. With the help of group theory we will study groups of polynomial permutations over rings of dual numbers over finite rings: their Sylow groups, their normal subgroups, and the projective limit of systems of such groups. In algebraic K-theory, eventually we want to determine how far matrices with determinant 1 over Int(Z) are away from being products of elementary matrices, and to study the structure of the special linear group of 2x2-matrices over Int(Z), where Int(Z) is the ring of integer-valued polynomials on the ring of integers. As a first step in this direction, we propose to show that the stable rank of Int(Z) is 2.