Endomorphism rings of toroidal solenoids

We study the endomorphism ring $End(G_A)$ of a subgroup $G_A$ of ${\mathbb{Q}}^n$ defined by a non-singular $n\times n$-matrix $A$ with integer entries. In the case when the characteristic polynomial of $A$ is irreducible and an extra assumption holds if $n$ is not prime, we show that $End(G_A)$ is commutative and can be identified with a subring of the number field generated by an eigenvalue of $A$. The obtained results can be applied to studying endomorphisms of associated toroidal solenoids and ${\mathbb{Z}}^n$-odometers. In particular, we build a connection between toroidal solenoids and $S$-integer dynamical systems, provide a formula for the number of periodic points of a toroidal solenoid endomorphism, and show that the linear representation group of a ${\mathbb{Z}}^n$-odometer is computable.