In abstract algebra, the endomorphisms of an abelian groupX form a ring. This ring is called the endomorphism ringX, denoted by End; the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a ring, with the zero map as additive identity and the identity map as multiplicative identity. The functions involved are restricted to what is defined as a homomorphism in the context, which depends upon the category of the object under consideration. The endomorphism ring consequently encodes several internal properties of the object. As the resulting object is often an algebra over some ring R, this may also be called the endomorphism algebra. An abelian group is the same thing as a module over the ring of integers, which is the initial ring. In a similar fashion, if R is any commutative ring, the endomorphism monoids of its modules form algebras over R by the same axioms and derivation. In particular, if R is a fieldF, its modules M are vector spacesV and their endomorphism rings are algebras over the fieldF. These observations are the starting point for enriched category theory, as the categories Ab, R-Mod and F-Vect have hom functors valued in the categories of Z-, R- and F-algebras, and are thus enriched in themselves.
Description
Let be an abelian group and we consider the group homomorphisms from A into A. Then addition of two such homomorphisms may be defined pointwise to produce another group homomorphism. Explicitly, given two such homomorphisms f and g, the sum of f and g is the homomorphism. Under this operation End is an abelian group. With the additional operation of composition of homomorphisms, End is a ring with multiplicative identity. This composition is explicitly. The multiplicative identity is the identity homomorphism on A. If the set A does not form an abelian group, then the above construction is not necessarily additive, as then the sum of two homomorphisms need not be a homomorphism. This set of endomorphisms is a canonical example of a near-ring that is not a ring.
Properties
Endomorphism rings always have additive and multiplicative identities, respectively the zero map and identity map.
A module is indecomposableif and only if its endomorphism ring does not contain any non-trivial idempotent elements. If the module is an injective module, then indecomposability is equivalent to the endomorphism ring being a local ring.
The endomorphism ring of a nonzero right uniserial module has either one or two maximal right ideals. If the module is Artinian, Noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring.
The endomorphism ring of an Artinian uniform module is a local ring.
If an R module is finitely generated and projective, then the endomorphism ring of the module and R share all Morita invariant properties. A fundamental result of Morita theory is that all rings equivalent to R arise as endomorphism rings of progenerators.
