We introduce a weakened form of regularity, the so called semiregularity, and we show that if every diagonal subalgebra of $\mathcal A \times \mathcal A$ is semiregular then $\mathcal A$ is congruence modular at 0.
The article studies the cubic mapping graph $\Gamma (n)$ of $\mathbb {Z}_n[{\rm i}]$, the ring of Gaussian integers modulo $n$. For each positive integer $n>1$, the number of fixed points and the in-degree of the elements $\overline 1$ and $\overline 0$ in $\Gamma (n)$ are found. Moreover, complete characterizations in terms of $n$ are given in which $\Gamma _{2}(n)$ is semiregular, where $\Gamma _{2}(n)$ is induced by all the zero-divisors of $\mathbb {Z}_n[{\rm i}]$.