# proof of Dedekind$-$Mertens lemma

Let $R$ be subring of the commutative ring $T$ and

 $f(X)=f_{0}+f_{1}X+\ldots+f_{m}X^{m}\quad\mbox{and}\quad g(X)=g_{0}+g_{1}X+% \ldots+g_{n}X^{n}$

be arbitrary polynomials in $T[X]$.  We will prove by induction on $n$ that the $R$-submodules of $T$ generated by the coefficients of the polynomials $f$, $g$, and $fg$ satisfy

 $\displaystyle M_{f}^{n+1}M_{g}\;=\;M_{f}^{n}M_{fg}$ (1)

where the product modules are generated by the products of their generators.

The generators of the right hand side of (1) belong obviously to the left hand side, whence only the containment

 $\displaystyle M_{f}^{n+1}M_{g}\;\subseteq\;M_{f}^{n}M_{fg}$ (2)

has to be proved.

Firstly, (2) is trivial in the case  $n=0$.  Let now  $n>0$.  Define

 $f_{j}\;:=\;0\quad\mbox{for}\quad j<0\quad\mbox{or}\quad j>m$

and let $G_{n}$ be the $R$-submodule of $T$ generated by $g_{0},g_{1},\ldots,g_{n-1}$.  We have

 $\sum_{i

where $h_{k}$ is the coefficient of $X^{k}$ of the polynomial $fg$, and thus by induction we can write

 $M_{f}^{n}G_{n}\;\subseteq\;M_{f}^{n-1}(M_{fg}+g_{n}M_{f})\;\subseteq\;M_{f}^{n% -1}M_{fg}+M_{f}^{n}g_{n}.$

This implies the containment

 $f_{i}M_{f}^{n}G_{n}\;\subseteq\;M_{f}^{n}M_{fg}+M_{f}^{n}f_{i}g_{n}$

for every $i$.  In addition, we have

 $f_{i}g_{n}\;\in\;M_{fg}+f_{i+1}G_{n}+M_{i+2}G_{n}+\ldots+f_{n}G_{n},$

whence

 $f_{i}M_{f}^{n}G_{n}\;\subseteq\;M_{f}^{n}M_{fg}+f_{i+1}M_{f}^{n}G_{n}+\ldots+f% _{n}M_{f}^{n}G_{n}.$

From this we infer that

 $f_{i}M_{f}^{n}G_{n}\;\subseteq\;M_{f}^{n}M_{fg}$

is true for each  $i$ $=$ $m$, $m\!-\!1,\,\ldots,\,0$.  Thus also (2) is true.

## References

• 1 J. Pahikkala: “Some formulae for multiplying and inverting ideals”.  – Ann. Univ. Turkuensis 183 (A) (1982).
• 2 J. Arnold & R. Gilmer: “On the contents of polynomials”.  – Proc. Amer. Math. Soc. 24 (1970).
• 3 T. Coquand: “A direct proof of Dedekind–Mertens lemma”. University of Gothenburg 2006. (Available http://www.cse.chalmers.se/ coquand/mertens.pdfhere.)
Title proof of Dedekind$-$Mertens lemma ProofOfDedekindMertensLemma 2013-12-15 20:29:20 2013-12-15 20:29:20 pahio (2872) pahio (2872) 5 pahio (2872) Proof msc 13A15 msc 13M10 msc 16D10 msc 16D25