Representation of rings 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)
 A: Exercise 2 in Chapter 1 of Krylov, Mikhalev, and Tuganbaev's Endomorphism Rings of Abelian Groups asks to show that $\mathbb{F}_p \times \mathbb{F}_p$ is not the endomorphism ring of any abelian group.  This is pretty clear: since $\mathbb{F}_p$ acts on the group $G$, as Kevin says $G$ must be an $\mathbb{F}_p$-vector space.
A: 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]

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 R. Gobel, B. Goldsmith.
Classifying E-algebras over Dedekind domains
Jnl. Algebra, Vol. 306, 2006, 566-575
