The endomorphisms of an abelian group form a ring under pointwise group operation and composition. Every ring is isomorphic to a subring of the endomorphism ring of some abelian group (left module over itself).


Is every ring isomorphic to the endomorphism ring of some abelian group? (not just a subring)



Representation of rings
Generally it's difficult to characterize rings isomorphic to an endomorphism ring of an abelian group. Interest in such problems was sparked by a problem given by Fuchs in his widely-read monograph *Abelian Groups*, cf. the excerpt below from the introduction to the [paper [1]](http://dx.doi.org/10.1016/j.jalgebra.2006.01.036)



> 
> The notion of an E-ring goes back to a
>  seminal paper of Schultz [20] written
>  in response to Problem 45 in the
>  well-known book `Abelian Groups' by
>  Laszlo Fuchs [11]. In this paper
>  Schultz distinguished between two
>  possibly different approaches, the
>  first we will continue to call an
>  E-ring, while the second we shall
>  refer to as a generalized E-ring. Thus
>  a ring R is said to be an E-ring if R
>  is isomorphic to the endomorphism ring
>  of its underlying additive group, R+,
>  via the mapping sending an element r
>  $\in$ R to the endomorphism given by
>  left multiplication by r, whilst R is
>  a generalized E-ring if some
>  isomorphism, not necessarily left
>  multiplication, exists between R and
>  its endomorphism ring End(R+). Since
>  right multiplication is always an
>  endomorphism, it is not difficult to
>  see that E-rings are necessarily
>  commutative. The existence of a
>  non-commutative generalized E-ring has
>  recently been established [15], and so
>  it follows that the class of
>  generalized E-rings is strictly larger
>  than the class of E-rings.
> 
> 
> Since Schultz's original paper there
>  has been a great deal of interest in
>  E-rings and some natural
>  generalizations, see e.g.
>  [1,2,4,6,8-10,17,19,21]. A notable
>  feature of much of this recent work
>  has been the use of so-called
>  realization theorems, whereby a
>  cotorsion-free ring is realized, using
>  combinatorial ideas derived from
>  Shelah's Black Box - see e.g. [7] for
>  details of this technique - as the
>  endomorphism ring of an Abelian group.
>  This present work arose from an
>  observation of the second author in
>  response to a question from the first
>  about the existence of generalized
>  E-algebras over the ring $J\_p$ of
>  p-adic integers; see [16] for further
>  details. A natural question which
>  arises, is to what extent is it
>  necessary for a ring to be
>  cotorsion-free in order to be a
>  generalized E-ring and the principal
>  objective of this work is to
>  characterize generalized E-rings
>  `modulo cotorsion-free groups.' The
>  characterization is quite elementary
>  but seems to have been overlooked
>  heretofore. It should be noted that
>  Bowshell and Schultz showed in [2]
>  that a reduced cotorsion E-ring has
>  the form $\prod\_{p \in U} {\mathbb
> > Z}(p^{k\_p}) \oplus \prod\_{p\in V} J\_p$
>  where $U,V$ are disjoint sets of
>  primes.
> 
> 
> 


[1](http://dx.doi.org/10.1016/j.jalgebra.2006.01.036) R. Gobel, B. Goldsmith.  

Classifying E-algebras over Dedekind domains  

Jnl. Algebra, Vol. 306, 2006, 566-575


