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The associative ring  $   \mathop{\rm End}  A =  \mathop{\rm Hom} ( A , A ) $
consisting of all morphisms of  $  A $
into itself, where  $  A $
is an object in some [[Additive category|additive category]]. The multiplication in  $   \mathop{\rm End}  A $
is composition of morphisms, and addition is the addition of morphisms defined by the axioms of the additive category. The identity morphism  $  1 _ {A} $
is the unit element of the ring  $   \mathop{\rm End}  A $. 
An element  $  \phi $
in  $   \mathop{\rm End}  A $
is invertible if and only if  $  \phi $
is an automorphism of the object  $  A $. 
If  $  A $
and  $  B $
are objects of an additive category  $  C $, 
then the group  $   \mathop{\rm Hom} ( A , B ) $
has the natural structure of a right module over  $   \mathop{\rm End}  A $
and of a left module over  $   \mathop{\rm End}  B $. 
Let  $  T :  C \rightarrow C _ {1} $
be a covariant (or contravariant) additive functor from an additive category  $  C $
into an additive category  $  C _ {1} $. 
Then for any object  $  A $
in  $  C $
the functor  $  T $
induces a natural homomorphism (or anti-homomorphism)  $   \mathop{\rm End}  A \rightarrow  \mathop{\rm End}  T ( A) $.

Let  $  C $
be the category of modules over a ring  $  R $. 
For an  $  R $-
module  $  A $
the ring  $   \mathop{\rm End}  A $
consists of all endomorphisms of the Abelian group  $  A $
that commute with multiplication by elements of  $  R $. 
The sum of two endomorphism  $  \phi $
and  $  \psi $
is defined by the formula

$$ 
( \phi + \psi ) ( a)  =  \phi ( a) +
\psi ( a) ,\  a \in A .
$$

If  $  R $
is commutative, then  $   \mathop{\rm End}  A $
has the natural structure of an  $  R $-
algebra. Many properties of the module  $  A $
can be characterized in terms of  $   \mathop{\rm End}  A $. 
For example,  $  A $
is an irreducible module if and only if  $   \mathop{\rm End}  A $
is a skew-field.

An arbitrary homomorphism  $  \pi $
of an associative ring  $  K $
into  $   \mathop{\rm End}  A $
is called a representation of the ring  $  K $(
by endomorphisms of the object  $  A $). 
If  $  K $
has a unit element, then one imposes the additional condition  $  \pi ( 1) = 1 _ {A} $. 
Any associative ring  $  K $
has a faithful representation in the endomorphism ring of a certain Abelian group  $  A $. 
If  $  K $, 
moreover, has a unit element, then  $  A $
can be chosen as the additive group of  $  K $
on which the elements of  $  K $
act by left multiplication. If  $  K $
has no unit element and  $  K _ {1} $
is obtained from  $  K $
by adjoining a unit to  $  K $
externally, then  $  A $
can be taken to be the additive group of  $  K _ {1} $.

In the case of an Abelian variety  $  X $
one considers, apart from the ring  $   \mathop{\rm End}  X $, 
which is a finitely-generated  $  \mathbf Z $-
module, the algebra of endomorphisms (algebra of complex multiplications)  $   \mathop{\rm End}  ^ {0}  X = \mathbf Q \otimes _ {\mathbf Z }   \mathop{\rm End}  X $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Faith,   "Algebra: rings, modules, and categories" , '''1–2''' , Springer  (1973–1976)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Mumford,   "Abelian varieties" , Oxford Univ. Press  (1974)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.T. Markov,   A.V. Mikhalev,   L.A. Skornyakov,   A.A. Tugaubaev,   "Endomorphism rings of modules and lattices of submodules"  ''J. Soviet Math.'' , '''31''' :  3  (1985)  pp. 3005–3051  ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''21'''  (1983)  pp. 183–254</TD></TR></table>