The classical ring of integer-valued polynomials Int(Z) consists of the polynomials in Q[X] that map Z into Z. We consider a generalization of integer-valued polynomials where elements of Q[X] act on sets such as rings of algebraic integers or the ring of n × n matrices with entries in Z. The collection of polynomials thus produced is a subring of Int(Z), and the principal question we consider is whether it is a Prüfer domain. This question is answered affirmatively for algebraic integers and negatively for matrices, although in the latter case Prüfer domains arise as the integral closures of the polynomial rings under consideration.

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Let D be an integrally closed domain with quotient field K. Let A be a torsion-free D-algebra that is finitely generated as a D-module. For every a in A we consider its minimal polynomial µa(X) ∈ D[X], i.e. the monic polynomial of least degree such that µa(a) = 0. The ring IntK (A) consists of polynomials in K[X] that send elements of A back to A under evaluation. If D has finite residue rings, we show that the integral closure of IntK(A) is the ring of polynomials in K[X] which map the roots in an algebraic closure of K of all the µa(X), a ∈ A, into elements that are integral over D. The result is obtained by identifying A with a D-subalgebra of the matrix algebra Mn(K) for some n and then considering polynomials which map a matrix to a matrix integral over D. We also obtain information about polynomially dense subsets of these rings of polynomials.

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We classify the maximal subrings of the ring of n × n matrices over a finite field, and show that these subrings may be divided into three types. We also describe all of the maximal subrings of a finite semisimple ring, and categorize them into two classes. As an application of these results, we calculate the covering number of a finite semisimple ring.