A group is called Hopfian  if it is not isomorphic to any of its proper factor groups, or, equivalently, any of its epimorphisms on itself is an isomorphism, i.e., an automorphism. This property was first proved by the Swiss mathematician H. Hopf for fundamental groups of Riemann surfaces. The results of the present paper concentrate around the problem of investigating general properties of Hopfian abelian groups and describing Hopfian groups in certain classes of abelian groups. Among the questions relating to Hopfian abelian groups, the study of the hopficity property in such a specific class of abelian groups as additive groups of rings occupies an important place. Additive groups of rings are one of the directions of research connecting the theory of abelian groups with the theory of rings. With regards to the methods of investigation and the nature of the results, this newly emerged direction, which appeared in the middle of the last century, is traditionally referred to the theory of abelian groups. When considering additive groups of particular classes of rings, some interesting examples of Hopfian abelian groups arise. The paper studies the hopficity in additive groups of $E$-rings (also called $E$-groups) and artinian rings. The work, in particular, proves that the additive group of an $E$-ring is Hopfian, and also gives a full description of how Hopfian additive groups of artinian rings are structured.