Vipul at 01:49, 4 July 2012

2012-07-04T01:49:31Z

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	==Definition==		==Definition==

	Let <math>R</math> be a [[commutative unital ring]]. The '''divided polynomial ring in one variable''' with indeterminate <math>x</math> over <math>R</math>, also called the '''free divided power algebra in one variable''', is defined as the ring obtained by adjoining formal symbols <~~mth~~>x^{(n)}</math> for all natural numbers <math>n</math> to <math>R</math>, subject to the following relations for all natural numbers <math>n</math> and all <math>i</math> with <math>0 < i < n</math>:		Let <math>R</math> be a [[commutative unital ring]]. The '''divided polynomial ring in one variable''' with indeterminate <math>x</math> over <math>R</math>, also called the '''free divided power algebra in one variable''', is defined as the ring obtained by adjoining formal symbols <math>x^{(n)}</math> for all natural numbers <math>n</math> to <math>R</math>, subject to the following relations for all natural numbers <math>n</math> and all <math>i</math> with <math>0 < i < n</math>:

	<math>x^{(i)}x^{(n-i)} = \binom{n}{i} x^{(n)}</math>		<math>x^{(i)}x^{(n-i)} = \binom{n}{i} x^{(n)}</math>

Vipul: Created page with "==Definition== Let $R$ be a commutative unital ring. The '''divided polynomial ring in one variable''' with indeterminate $x$ over $R$ , a..."

2012-07-04T01:49:11Z

Created page with "==Definition== Let <math>R</math> be a commutative unital ring. The '''divided polynomial ring in one variable''' with indeterminate <math>x</math> over <math>R</math>, a..."

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==Definition==

Let <math>R</math> be a [[commutative unital ring]]. The '''divided polynomial ring in one variable''' with indeterminate <math>x</math> over <math>R</math>, also called the '''free divided power algebra in one variable''', is defined as the ring obtained by adjoining formal symbols <mth>x^{(n)}</math> for all natural numbers <math>n</math> to <math>R</math>, subject to the following relations for all natural numbers <math>n</math> and all <math>i</math> with <math>0 < i < n</math>:

<math>x^{(i)}x^{(n-i)} = \binom{n}{i} x^{(n)}</math>

We can additionally set <math>x^{(0)} = 1</math> (so that the above becomes true with <math>0 \le i \le n</math>) and we denote <math>x^{(1)}</math> by <math>x</math>.

==Particular cases==

* In the case that <math>R</math> is a <math>\mathbb{Q}</math>-algebra, the divided polynomial ring is the same as <math>R[x]</math>, and the element <math>x^{(n)}</math> is identified with <math>x^n/n!</math>.
* In case the [[characteristic]] of <math>R</math> is zero, we can realize the divided polynomial ring as an intermediate subring between <math>R[x]</math> and <math>L[x]</math>, where <math>L</math> is the [[localization at a multiplicatively closed subset|localization]] of <math>R</math> at the multiplicatively closed subset of nonzero integers. Explicitly, <math>x^{(n)} = x^n/n!</math>, which makes sense inside <math>L[x]</math>.

==Related notions==

* [[Ring generated by binomial polynomials]]
* [[Ring of integer-valued polynomials]]

Divided polynomial ring - Revision history

Vipul at 01:49, 4 July 2012

Vipul: Created page with "==Definition== Let R be a commutative unital ring. The '''divided polynomial ring in one variable''' with indeterminate x over R, a..."

Vipul: Created page with "==Definition== Let $R$ be a commutative unital ring. The '''divided polynomial ring in one variable''' with indeterminate $x$ over $R$ , a..."