Integer-valued polynomials on algebras

Abstract

Let D be a domain with quotient field K and A a D-algebra. We call a polynomial with coefficients in K that maps every element of A to an element of A "integer-valued on A". For commutative A we also consider integer-valued polynomials in several variables. For an arbitrary domain D and I an arbitrary ideal of D we show I-adic continuity of integer-valued polynomials on A. For Noetherian one-dimensional D, we determine the spectrum and Krull dimension of the ring Int_D(A) of integer-valued polynomials on A. We do the same for the ring of polynomials with coefficients in M_n(K), the K-algebra of n x n matrices, that map every matrix in M_n(D) to a matrix in M_n(D).

17 pages; a glitch in the published version (J.Algebra 373 (2013) 414-425) has been corrected in this post-preprint, namely, in Prop. 6.2 and Thm. 6.3, the assumption "zero Jacobson radical" needs to be replaced by the stronger assumption "intersection of maximal ideals of finite index is zero"

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• Non-triviality Conditions for Integer-valued Polynomial Rings on Algebras
• Null ideals of matrices over residue class rings of principal ideal domains
• Decomposition of integer-valued polynomial algebras
• Polynomial Dedekind domains with finite residue fields of prime characteristic
• A Survey on Fixed Divisors
• Polynomial functions on non-commutative rings - a link between ringsets and null-ideal sets