On ranges of polynomials in the ring $M_2(\mathbb Z/8\mathbb Z)$

The main result of this article is the following: a subset $A$ of $2\times2$ matrices over the ring $\mathbb Z/8\mathbb Z$ is the range of a polynomial in noncommuting indeterminates with coefficients in $\mathbb Z/8\mathbb Z$ and without constant term if and only if $A$ contains 0 and is selfsimilar, that is $\alpha A\alpha^{-1}\subseteq A$ for each invertible $2\times2$ matrix $\alpha$.