Finiteness properties of matrix representations.(English)Zbl 0604.20027

Let $$\Gamma_ n={\mathbb{Z}}[x_{ij}:$$ $$i,j=1,...,n]$$ be the polynomial ring in $$n^ 2$$ variables with integer coefficients. For each polynomial $$\phi \in \Gamma_ n$$ there is a map of $$M_ n(A)$$, of the set of $$n\times n$$ matrices over the commutative ring A with 1, into A, where to each matrix $$[a_{ij}]$$ there corresponds the element $$\phi (a_{ij})$$ of A. But also to each square $$n\times n$$ matrix there is a unique ring homomorphism of $$\Gamma_ n$$ into A taking each variable $$x_{ij}$$ onto the element $$a_{ij}$$. So if $$\Phi$$ is a subset of $$\Gamma_ n$$, then $$\Phi$$ is a collection of functions of $$M_ n(A)$$ to A and a subset X of $$M_ n(A)$$ is said to satisfy $$\Phi$$ if all the functions in $$\Phi$$ are 0 on X. For a commutative ring A the set $$M_ n(A)$$ is a monoid under matrix multiplication.
Now the main result of this paper can be stated as follows: Let S be a finitely generated monoid. Let n be a positive integer, let $$\Phi \subset \Gamma_ n$$ and $$P\subset S$$. Then there exists a finite subset $$Q\subset P$$, such that, for every commutative ring A and every monoid homomorphism $$\mu$$ : $$S\to M_ n(A)$$ it is true that $$\mu$$ (P) satisfies $$\Phi$$ if and only if $$\mu$$ (Q) satisfies $$\Phi$$. The proof is based on an idea of V. S. Guba for the proof of the Ehrenfeucht Conjecture which uses the Hilbert Basis Theorem. Among the corollaries the author gets the proof of Ehrenfeucht’s conjecture on the equalizers of two monoid homomorphisms and, using Gersten’s fixed point theorem, the proof that the group of periodic points of an automorphism of a finitely generated free group is finitely generated.