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(RR) = 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.
Read more about this topic: Endomorphism Ring
Famous quotes containing the word examples:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (16881744)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (16701733)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (15331592)
Related Subjects
Related Phrases
Related Words