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When D is an integral domain with field of fractions K, the ring Int(D) = {f (x) ∈ K[x] | f (D) ⊆ D} of integer-valued polynomials over D has been extensively studied. We will extend the integer-valued polynomial construction to certain noncommutative rings. Specifically, let i, j, and k be the standard quaternion units satisfying the relations i 2 = j 2 =… (More)

The classical ring of integer-valued polynomials Int(Z) consists of the poly-nomials 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),… (More)

Given a finite (associative, unital) ring R, let K(R) denote the set of polynomials in R[x] that send each element of R to 0 under evaluation. We study K(R) and its elements. We conjecture that K(R) is a two-sided ideal of R[x] for any finite ring R, and prove the conjecture for several classes of finite rings (including commutative rings, semisimple rings,… (More)

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