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.