Endomorphism ring

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. 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

[1]	C. Faith, "Algebra: rings, modules, and categories" , 1–2 , Springer (1973–1976)
[2]	D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974)
[3]	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