Gapplet

Given a nonzero Gaussian integer a+bi Z+iZ, the structure of the residue class ring R = Z[i]/(a+bi) is determined.

Input a+bi, e.g., 2+3*I  (the number i is input as I)
Analyze the residue class ring R

The ring R is finite, because m = (a-bi)(a+bi) belongs to the ideal (a+bi)Z[i], so that R is a residue class ring of Z[i]/(m) = Z/mZ+iZ/mZ, which has clearly m² elements.