  In this paper, we explore some properties of Galois hulls of cyclic serial
codes over a chain ring and we devise an algorithm for computing all the
possible parameters of the Euclidean hulls of that codes. We also establish the
average $p^r$-dimension of the Euclidean hull, where $\mathbb{F}_{p^r}$ is the
residue field of $R$, and we provide some results of its relative growth.
