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 $f g$ 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<n}f_{k-i}g_{i}\;=\;h_{k}-f_{k-n}g_{n}\;\in\;M_{fg}+g_{n}M_{f}

where $h_{k}$ is the coefficient of $X^{k}$ of the polynomial $f g$ , 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
Canonical name	ProofOfDedekindMertensLemma
Date of creation	2013-12-15 20:29:20
Last modified on	2013-12-15 20:29:20
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	5
Author	pahio (2872)
Entry type	Proof
Classification	msc 13A15
Classification	msc 13M10
Classification	msc 16D10
Classification	msc 16D25

proof of Dedekind-Mertens lemma

References

proof of Dedekind $-$ Mertens lemma