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2008-05-12T16:26:33Z

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Vipul at 17:21, 7 January 2008

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==Definition==

Let <math>R</math> be a [[commutative unital ring]], and <math>R[x]</math> be the [[polynomial ring]] in one variable over <math>R</math>. The '''leading coefficient map''' is a map from <math>R[x]</math> to <math>R</math>, which sends any polynomial to its leading coefficient, i.e. the coefficient of the highest power of the indeterminate which has a nonzero coefficient.

==Facts==

The leading coefficient map is not a ring homomorphism in general, but it has the following properties:

* Given a constant polynomial, its leading coefficient is the constant value. Hence the composite of the inclusion <math>R \to R[x]</math> and the leading coefficient map is the identity map (one way).
* When the ring is an [[integral domain]], the leading coefficient map is a homomorphism of the multiplicative monoids. In other words, the leading coefficient of the product of two polynomials is the product of their leading coefficients (this can fail for non-integral domains because the product of the leading coefficients could be zero).
* If two polynomials, and their sum, all have the same degree, then the leading coefficient of the sum is the sum of the leading coefficients.
* The leading coefficient map sends ideals in <math>R[x]</math> to ideals in <math>R</math>. Further, if the image of an ideal under the leading coefficient map is finitely generated, then so is the original ideal. {{further|[[image under leading coefficient is finitely generated implies finitely generated]]}}

Leading coefficient map - Revision history

Vipul: 1 revision

Vipul at 17:21, 7 January 2008