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:
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“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 (1670–1733)