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).Comment: 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

A Survey on Fixed Divisors

“…In recent years, several prominent mathematicians have studied the ring of polynomials in M m (K) [x] which maps M m (R) back to this ring, generally denoted by Int(M m (R)). For various interesting results about this ring, we refer to [46], [45], [48], [52], [51], [62], [71], [84], [85], [88], [89], [120]. For a survey on Int(M m (R)), the reader may consult [49] and [123].…”

Polynomial functions on upper triangular matrix algebras

“…ij is the (i, j)-th entry of the k-th power of a generic upper triangular n × n matrix (with n ≥ i, j) whose (i, j)-th entry is x ij when i ≤ j and zero otherwise. Again, note that p 24 is the sequence of entries in position (2,4) in the powers G 0 , G, G 2 , G 3 , . .…”

Polynomial functions on subsets of non-commutative rings — a link between ringsets and null-ideal sets

“…is the ring of n × n matrices over D. Here, Int l Mn(K) (M n (D)) coincides with Int r Mn(K) (M n (D)) (shown to be a ring by Werner [12]), and is canonically isomorphic to M n (Int K (M n (D))) [4]. The algebras for which Int B (A) ≃ Int K (A) ⊗ D A thus holds have been characterized by Peruginelli and Werner [10].…”

“…In the theory of polynomial mappings on commutative rings, there are two notable subtopics, namely, polynomial functions on finite rings, and rings of integervalued polynomials. Here, we are concerned with generalizations of these two topics to polynomial mappings on non-commutative rings, as proposed by Loper and Werner [8], and developed further by Werner [11][12][13][14], Peruginelli [9,10], and the present author [3][4][5], among others. More particularly, we will investigate connections between null ideals of polynomials on finite non-commutative rings and integer-valued polynomials on non-commutative rings.…”

“…When we talk about polynomial functions on a non-commutative ring R, we mean functions induced by elements of the usual polynomial ring R[x] whose indeterminate x commutes with the elements of R. Non-commutative rings R for which polynomial functions have been studied include rings of quaternions [7,11,14], and matrix algebras [4,5,12].…”