On images of polynomials in finite matrix rings

We study the images of polynomials in non-commuting indeterminates in the ring of $2\times2$ matrices over a Galois ring. The main result: a set of $2\times2$ matrices over a Galois ring whose radical has nilpotency index 2, is an image of a polynomial with zero constant term if and only if it contains 0 and is self-conjugate.