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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 and $f,\,g\in R[x]$, the Gauss lemma I
implies ^{1}^{1}In a UFD, one can use as contents of $f$ and $g$ the greatest 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 *Gaussian polynomial*, 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 Gaussian polynomials.

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 the
*Hilfssatz von Dedekind–Mertens*, i.e. the
Dedekind–Mertens lemma. A generalised form of it is in the
entry
product of finitely generated ideals.

# 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).

## Mathematics Subject Classification

11C08*no label found*

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