On $\Sigma$-subsets of naturals over abelian groups

We obtain conditions for the $\Sigma$-definability of a subset of the set of naturals in the hereditarily finite admissible set over a model and for the computability of a family of such subsets. We prove that: for each $e$-ideal $I$ there exists a torsion-free abelian group $A$ such that the family of $e$-degrees of $\Sigma$-subsets of $\omega$ in $\mathbb{HF}(A)$ coincides with $I$ there exists a completely reducible torsion-free abelian group in the hereditarily finite admissible set over which there exists no universal $\Sigma$-function; for each principal $e$-ideal $I$ there exists a periodic abelian group $A$ such that the family of $e$-degrees of $\Sigma$-subsets of $\omega$ in $\mathbb{HF}(A)$ coincides with $I$.