Ring generated by binomial polynomials: Difference between revisions

Latest revision as of 01:31, 4 July 2012

This is a variation of polynomial ring
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Definition

Let $R$ be a commutative unital ring of characteristic zero. Let $K$ be the ring obtained by localizing $R$ at the multiplicative subset of nonzero integers. Then, the ring generated by binomial polynomials over $R$ is the subring of $K[x]$ comprising all $R$ -linear combinations of the polynomials:

${\binom {x}{r}}={\frac {x(x-1)(x-2)\dots (x-r+1)}{r!}}$ .

where $r\geq 0$ (for $r=0$ , this is the constant polynomial $1$ ).

Equivalently, it is the tensor product with $R$ of the ring generated by binomial polynomials over the rational integers, i.e., the ring generated by binomial polynomials over $\mathbb {Z}$ .

Equivalently, it is the ring $\operatorname {Int} (\mathbb {Z} ,R)$ : the ring of all polynomials $f\in K[x]$ such that $f(\mathbb {Z} )\subseteq R$ .

Facts

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+{{variation of|polynomial ring}}
 ==Definition==
-Let <math>R</math> be a [[commutative unital ring]] of [[characteristic of a ring|characteristic]] zero. Let <math>K</math> be the [[quotient ring]] of <math>R</math> by the multiplicative subset of nonzero integers. Then, the '''ring generated by binomial polynomials''' over <math>R</math> is the subring of <math>K[x]</math> comprising all <math>R</math>-linear combinations of the polynomials:
+Let <math>R</math> be a [[commutative unital ring]] of [[characteristic of a ring|characteristic]] zero. Let <math>K</math> be the ring obtained by [[localization at a multiplicatively closed subset|localizing]] <math>R</math> at the multiplicative subset of nonzero integers. Then, the '''ring generated by binomial polynomials''' over <math>R</math> is the subring of <math>K[x]</math> comprising all <math>R</math>-linear combinations of the polynomials:
 <math>\binom{x}{r} = \frac{x(x-1)(x-2) \dots (x - r + 1)}{r!}</math>.
@@ Line 10: / Line 11: @@
 Equivalently, it is the ring <math>\operatorname{Int}(\mathbb{Z},R)</math>: the ring of all polynomials <math>f \in K[x]</math> such that <math>f(\mathbb{Z}) \subseteq R</math>.
+==Facts==
+* [[Ring of integer-valued polynomials is contained in ring generated by binomial polynomials]]
+* [[Ring of integer-valued polynomials over rational integers equals ring generated by binomial polynomials]]