content of polynomial

The content of a polynomial $f$ may be defined in any polynomial ring $R[x]$ over a commutative ring $R$ as the ideal of $R$ generated by the coefficients of the polynomial. It is denoted by $\\operatorname{cont}(f)$ or $c(f)$ . Coefficient module is a little more general concept.

If $R$ is a unique factorisation domain (http://planetmath.org/UFD) and $f,\\,g\\in R[x]$ , the Gauss lemma Iimplies ¹¹In a UFD, one can use as contents of $f$ and $g$ the http://planetmath.org/node/5800greatest common divisors $a$ and $b$ of the coefficients of these polynomials, when one has $f(x)=af_{1}(x)$ , $g(x)=bg_{1}(x)$ with $f_{1}(x)$ and $g_{1}(x)$ primitive polynomials. Then $f(x)g(x)=abf_{1}(x)g_{1}(x)$ , and since also $f_{1}g_{1}$ is a primitive polynomial, we see that $c(fg)=ab=c(f)c(g)$ . that

\\displaystyle c(fg)\\;=\\;c(f)c(g).

(1)

For an arbitrary commutative ring $R$ , there is only the containment

\\displaystyle c(fg)\\;\\subseteq\\;c(f)c(g)

(2)

(cf. product of finitely generated ideals). The ideal $c(fg)$ is called the Gaussian ideal of the polynomials $f$ and $g$ . The polynomial $f$ in $R[x]$ is a , if (2) becomes the equality (1) for all polynomials $g$ in the ring $R[x]$ . The ring $R$ is a Gaussian ring, if all polynomials in $R[x]$ are .

It’s quite interessant, that the equation (1) multiplied by the power $[c(f)]^{n}$ , where $n$ is the degree of the other polynomial $g$ , however is true in any commutative ring $R$ , thus replacing the containment (2):

\\displaystyle[c(f)]^{n}c(fg)\\;=\\;[c(f)]^{n+1}c(g).

(3)

This result is called theHilfssatz von Dedekind–Mertens, i.e. theDedekind–Mertens lemma. A generalised form of it is in theentryproduct of finitely generated ideals (http://planetmath.org/ProductOfFinitelyGeneratedIdeals).

References

1 Alberto Corso & Sarah Glaz: “Gaussian ideals and the Dedekind–Mertens lemma” in Jürgen Herzog & Gaetana Restuccia (eds.): Geometric and combinatorial aspects of commutative algebra. Marcel Dekker Inc., New York (2001).