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

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 $\alpha$ so that the rings of Gaussian integers modulo $\alpha$ satisfy the diagonal condition.