  This paper is devoted to tetrahedron maps, which are set-theoretical
solutions to the Zamolodchikov tetrahedron equation. We construct a family of
tetrahedron maps on associative rings. We show that matrix tetrahedron maps
presented in [arXiv:2110.05998] are a particular case of our construction. This
provides an algebraic explanation of the fact that the matrix maps from
[arXiv:2110.05998] satisfy the tetrahedron equation. Also, Liouville
integrability is established for some of the constructed maps.
