Motivation: for any ring $R$ there is the natural monomorphism $\mathrm{in} \colon R \to \mathrm{End}(R_{add}): r \mapsto (x \mapsto rx)$, where $R_{add}$ is an additive abelian group ( rings are assumed to be associative with identity, but not necessarily commutative). So a ring is exactly an Abelian group with a distinguished subgroup of its endomorphisms (and a fixed bijection between elements and distinguished endomorphisms). Some rings are "complete" in the sense that they "contain" all endomorphisms of the underlying abelian group. For example, $\mathrm{in}$ is an isomorphism for $R = \mathbb{Z}, \mathbb{Q}, \mathbb{Z}_n$.


1. What is known about the classification of Abelian groups $A$ such that there is an isomorphism between the Abelian groups $\mathrm{End}(A)$ and $A$?

Each such isomorphism gives some "complete ring" structure on $A$.

2. What is known about the uniqueness (up to isomorphism) of the "complete ring" structure on an Abelian group?

I'm interested in the answers to these questions, with any additional assumptions that seem natural to you. I am especially interested in the answers for commutative rings.