Let \(\theta\) be an algebraic integer over \(\mathbb{Q}\) and \(A\subset \mathbb{Z}[\theta]\) be a complete residue system \(\text{mod } \theta\). The authors consider expansions of the elements \(\alpha\in \mathbb{Z}[\theta]\) in powers of \(\theta\) with coefficients \(\alpha_ k\in A\) defined by \(\alpha_ k= \alpha_{k+1}\theta+b_ k\), where \(b_ k\in A\) and they prove some results concerning the set \(S\) of purely periodic expansions. If \(S=\{0\}\), then \((\theta,A)\) is called a number system.