$\Sigma$-uniform structures and $\Sigma$-functions. II

We construct a family of $\Sigma$-uniform Abelian groups and a family of $\Sigma$-uniform rings. Conditions are specified that are necessary and sufficient for a universal $\Sigma$-function to exist in a hereditarily finite admissible set over structures in these families. It is proved that there is a set $S$ of primes such that no universal $\Sigma$-function exists in hereditarily finite admissible sets $\mathbb{HF}(G)$ and $\mathbb{HF}(K)$, where $G=\oplus\{Z_p\mid p\in S\}$ is a group, $Z_p$ is a cyclic group of order $p$, $K=\oplus\{F_p\mid p\in S\}$ is a ring, and $F_p$ is a prime field of characteristic $p$.