A non-abelian 2-group whose endomorphisms generate a ring, and other examples of E-groups
Groups for which the distributively generated near-ring generated by the endomorphisms is in fact a ring are known as E-groups and are discussed in (3). R. Faudree in (1) has given the only published examples of non-abelian E-groups by presenting defining relations for a family of p–groups. However, as shown in (3), Faudree's group does not have the desired property when p = 2.
An Interesting Orbit-Induced Partition of the Class of all Groups
Given that the set of endomorphisms of a group is contained in the set of distributive elements of its endomorphism near-ring, which, in turn, is contained in the endomorphism near-ring, we show that the class of all groups is partitioned into four nonempty subclasses when all combinations of these inclusions, proper or non-proper, are considered. Furthermore, a characterization of each subclass is given in terms of the orbits of the underlying group.
On distributively generated near-rings
The following theorems in ring theory are well-known: 1. Let R be a ring. If e is a unique left identity, then e is also a right identity. 2. If R is a ring with more than one element such that aR = R for every nonzero element a ε R, then R is a division ring. 3. A ring R with identity e ≠ 0 is a division ring if and only if it has no proper right ideals.