E-groups and E-rings

An associative ring $R$ is called an $E$-ring if the canonical homomorphism $R\cong \textsf{E}(R^+)$ is an isomorphism. Additive groups of $E$-rings are called $E$-groups. In other words, an Abelian group $A$ is an $E$-group if and only if $A\cong \operatorname{End} A$ and the endomorphism ring $\textsf{E}(A)$ is commutative. In this paper, we give a survey of the main results on $E$-groups and $E$-rings and also consider some of their generalizations: $\mathcal{E}$-closed groups, $T$-rings, $A$-rings, the groups admitting only commutative multiplications, etc.