Non-triviality conditions for integer-valued polynomial rings on algebras
نویسندگان
چکیده
منابع مشابه
Integer-valued Polynomials on Algebras
Let D be a domain with quotient field K and A a D-algebra. A polynomial with coefficients in K that maps every element of A to an element of A is called 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-d...
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We compare several different concepts of integer-valued polynomials on algebras and collect the few results and many open questions to be found in the literature. (2000 Math. Subj. Classification: Primary 13F20; Secondary 16S50, 13B25, 13J10, 11C08, 11C20)
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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 integervalued 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...
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Every integer is either even or odd, so we know that the polynomial f(x) = x(x− 1) 2 is integervalued on the integers, even though its coefficients are not in Z. Similarly, since every binomial coefficient ( k n ) is an integer, the polynomial ( x n ) = x(x− 1)...(x− n+ 1) n! must also be integervalued. These polynomials were used for polynomial interpolation as far back as the 17 century. Inte...
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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 = j = −1 and ij = k = −ji, and define ZQ := {a+bi+...
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ژورنال
عنوان ژورنال: Monatshefte für Mathematik
سال: 2016
ISSN: 0026-9255,1436-5081
DOI: 10.1007/s00605-016-0951-8