Content of a polynomial

Let $R$ be a commutative ring with $1$, and let $f(x)=\sum_{i=0}^n a_i x^i\in R[x]$. The content of $f$ is the ideal

i.e. the ideal generated by the coefficients of $f$.

When $R$ is a gcd domain (e.g. a UFD ), the content corresponds (up to associates ) to the gcd of the coefficients. The notion is central to Gauss-type results about how factorization in $R[x]$ relates to factorization in $\mathrm{Frac}(R)[x]$.

Examples:

• In $\mathbb{Z}[x]$, for $f=6x^2+15x$ one has $c(f)=(3)$.
• In $\mathbb{Z}[x]$, for $g=x^2+2$ one has $c(g)=(1)$.
• For the zero polynomial $0\in R[x]$, $c(0)=(0)$.