# 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