Diagonal Condition in Multiplication Table of $\displaystyle {\, \mathbb{Z} [i] / (α) }$

Abstract

Multiplication table of a ring with identity 1 is said to have the diagonal condition if 1s occur only on the main diagonal. In this paper, we study the diagonal condition in the ring of Gaussian integers $\mathbb{Z}[i]$. Furthermore, we also find the Gaussian integers $α$ so that the rings of Gaussian integers modulo $α$ satisfy the diagonal condition.