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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.
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(A) is an abelian group. With the additional operation of composition of homomorphisms, End(A) 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.
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.
If an R module is finitely generated and projective (that is, a progenerator), 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.
For any abelian group , , since any matrix in carries a natural homomorphism structure of as follows:
One can use this isomorphism to construct a lot of non-commutative endomorphism rings. For example: , since .
Also, when is a field, there is a canonical isomorphism , so , that is, the endomorphism ring of a -vector space is identified with the ring of n-by-n matrices with entries in . More generally, the endomorphism algebra of the free module is naturally -by- matrices with entries in the ring .
As a particular example of the last point, for any ring R with unity, , where the elements of R act on R by left multiplication.