Content of a polynomial

Definition

Let $R$ be a commutative unital ring and $f\in R[x]$ be a polynomial. The content of $f$ is defined as the ideal of $R$ generated by the coefficients of $f$.

In the case where $R$ is a gcd domain (for instance, where $R$ is a Bezout domain or a unique factorization domain), the content of $f$ is defined as the greatest common divisor of all the coefficients of $f$. Note that the greatest common divisor is completely determined (upto associates) by the ideal generated by the coefficients: it is the generator of the smallest principal ideal containing that ideal.

In the case that $R$ is a Bezout domain, the content of $f$ is in fact the generator of the ideal generated by all the coefficients.

Related notions

A primitive polynomial over a ring is a polynomial such that the ideal generated by the coefficients is not contained in any proper principal ideal. Equivalently, the greatest common divisor of the coefficients is $1$.