Subalgebras of Matrix Algebras Generated by Companion Matrices

Abstract

Let $f,g\in Z[X]$ be monic polynomials of degree $n$ and let $C,D\in M_n(Z)$ be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra $Z< C,D>$ to be a sublattice of finite index in the full integral lattice $M_n(Z)$, in which case we compute the exact value of this index in terms of the resultant of $f$ and $g$. If $R$ is a commutative ring with identity we determine when $R< C,D>=M_n(R)$, in which case a presentation for $M_n(R)$ in terms of $C$ and $D$ is given.