AE-rings

E-rings are a well known notion in the theory of abelian groups. They are those rings R such that End (R⁺), the ring of endomorphisms of the additive group of R, is as small as possible, i.e. End (R⁺)=Rℓ, where Rℓ={x↦ax:a∈R}(. We generalize the notion of E-rings by calling a ring R an almost- E-ring, or AE-ring for short, if End (R⁺) is a radical extension of Rℓ, i.e. for each φ∈End (R⁺) there is some natural number n such that φⁿ∈Rℓ. We will show that this notion does not lead to a new class of rings. It turns out that all AE-rings are actually E-rings. Our proof utilizes Herstein’s Hypercenter Theorem.