The Noetherian Property in Rings of Integer-Valued Polynomials
نویسندگان
چکیده
منابع مشابه
Generalized Rings of Integer-valued Polynomials
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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Let D be an integral domain and E = {e1, . . . , ek} a finite nonempty subset of D. Then Int(E,D) has the strong two-generator property if and only if D is a Bezout domain. If D is a Dedekind domain which is not a principal ideal domain, then we characterize which elements of Int(E, D) are strong two-generators. Let D be an integral domain with quotient field K and E ⊆ D a subset of D. We let I...
متن کاملInteger-valued Polynomials
Let R be a Krull ring with quotient field K and a1, . . . , an in R. If and only if the ai are pairwise incongruent mod every height 1 prime ideal of infinite index in R does there exist for all values b1, . . . , bn in R an interpolating integer-valued polynomial, i.e., an f ∈ K[x] with f(ai) = bi and f(R) ⊆ R. If S is an infinite subring of a discrete valuation ring Rv with quotient field K a...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 1993
ISSN: 0002-9947
DOI: 10.2307/2154451