Leading coefficient map

Definition

Let $R$ be a commutative unital ring, and $R[x]$ be the polynomial ring in one variable over $R$. The leading coefficient map is a map from $R[x]$ to $R$, 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 $R\to R[x]$ 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 $R[x]$ to ideals in $R$. Further, if the image of an ideal under the leading coefficient map is finitely generated, then so is the original ideal. Further information: image under leading coefficient is finitely generated implies finitely generated