# Endomorphism ring

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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. 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
How to Cite This Entry:
Endomorphism ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Endomorphism_ring&oldid=46821
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article