Elementary ideals are independent of the presentation

Given: A commutative ring $R$, a finitely presented $R$-module $M$ and an integer $k\ge 0$. No choice principle is used.

For a presentation $R^m\stackrel{A}{\to }{R}^n\to M\to 0$ with $m,n$ finite, $E_k(M)$ is the ideal generated by all $(n-k)\times (n-k)$ minors of $A$, with $E_k(M)=R$ for $k\ge n$ and $E_k(M)=0$ for $k<n-m$ (Elementary ideals of a finitely presented module).

For a finitely generated module $N$ presented as $R^{(J)}\stackrel{\phi }{\to }{R}^n\to N\to 0$ with $n$ finite and $J$ arbitrary, the ideal $I_{n-k}(\phi )$ generated by the $(n-k)\times (n-k)$ minors depends only on $N$ and the fixed integer $k$, not on the presentation; it is written ${\mathrm{Fitt}}_k(N)$. The conventions are $I_r=R$ for $r\le 0$ and $I_r=0$ when there are no $r\times r$ minors. In particular the minor size changes when the number of presentation generators changes. Fitting ideals are compatible with base change (Fitting ideals do not depend on a presentation).

A finitely presented module $M$ is finitely generated and admits a presentation $R^m\to R^n\to M\to 0$ with $m,n$ finite; the cokernel of the presentation map is $M$ (Finitely presented modules and finitely presented algebras, Module homomorphism and isomorphism, kernel, image and cokernel, Generated submodule, cyclic and finitely generated modules, module basis and free module).

An isomorphism $u:M\to M^{\mathrm{\prime }}$ of $R$-modules carries a presentation $R^m\to R^n\to M\to 0$ to the presentation $R^m\to R^n\stackrel{u\circ \beta }{\to }{M}^{\mathrm{\prime }}\to 0$ with the same presentation matrix; hence isomorphic modules admit presentations with identical matrices. [F3, given]