The factor rings of the ordinary integers $\mathbb Z$ are the well-known residual classes $\mathbb Z_n$. For the Gaussian integers $\mathbb Z[i]$ the factor rings are studied in
1) J. T. Cross, The Euler ϕ-function in the Gaussian integers, Amer. Math. Monthly 90 (1983) 518–528.
2) Dresden, Greg(1-WLEE); Dymàček, Wayne M.(1-WLEE) Finding factors of factor rings over the Gaussian integers. Amer. Math. Monthly 112 (2005), no. 7, 602–611.
There are articles on the factor rings of the ring of integers of an algebraic number field?