Endomorphism Ring - Examples

Examples

• In the category of R modules the endomorphism ring of an R module M will only use the R module homomorphisms, which are typically a proper subset of the abelian group homomorphisms. When M is a finitely generated projective module, the endomorphism ring is central to Morita equivalence of module categories.
• If K is a field and we consider the K-vector space Kn, then the endomorphism ring of Kn which consists of all K-linear maps from Kn to Kn. After a basis for the vector space is chosen, this ring is naturally identified with the ring of n-by-n matrices with entries in K. More generally, the endomorphism algebra of the free module is naturally n-by-n matrices with entries in R.
• As a particular example of the last point, for any ring R with unity, End(R_R) = R, where the elements of R act on R by left multiplication.
• In general, endomorphism rings can be defined for the objects of any preadditive category.