An upper cardinal bound on absolute E-rings

We show that for every abelian group $A$ of cardinality $\ge\kappa(\omega)$ there exists a generic extension of the universe, where $A$ is countable with $2^{\aleph_0}$ injective endomorphisms. As an immediate consequence of this result there are no absolute E-rings of cardinality $\ge \kappa(\omega)$. This paper does not require any specific prior knowledge of forcing or model theory and can be considered accessible also for graduate students.