Digit systems over commutative rings

Let $\text{E}$ be a commutative ring with identity and $P\in \text{E}[x]$ be a polynomial. In the present paper we consider digit representations in the residue class ring $\text{E}[x]\mathrm{/}(P)$. In particular, we are interested in the question whether each $A\in \text{E}[x]\mathrm{/}(P)$ can be represented modulo $P$ in the form $e_0+e_1 X + \cdots + e_h X^h$,wherethe$e_i\in\E[x]/(P)$ are taken from a fixed finite set of digits. This general concept generalises both canonical number systems and digit systems over finite fields. Due to the fact that we do not assume that $0$ is an element of the digit set and that $P$ need not be monic, several new phenomena occur in this context.