Description
Let A be an abelian group and f and g be two group homomorphisms from A into itself. Then the functions may be added pointwise to produce a group homomorphism. Under this operation End(A) is an Abelian group. With the additional operation of function composition, End(A) is a ring with multiplicative identity. The multiplicative identity is the identity function on A.
If the set A does not form an Abelian group, then the above construction does not result in the set of endomorphisms being an additive group, as the sum of two homomorphisms need not be a homomorphism in that case. This set of endomorphisms is a canonical example of a near-ring which is not a ring.
Read more about this topic: Endomorphism Ring
Famous quotes containing the word description:
“The Sage of Toronto ... spent several decades marveling at the numerous freedoms created by a “global village” instantly and effortlessly accessible to all. Villages, unlike towns, have always been ruled by conformism, isolation, petty surveillance, boredom and repetitive malicious gossip about the same families. Which is a precise enough description of the global spectacle’s present vulgarity.”
—Guy Debord (b. 1931)
“The great object in life is Sensation—to feel that we exist, even though in pain; it is this “craving void” which drives us to gaming, to battle, to travel, to intemperate but keenly felt pursuits of every description whose principal attraction is the agitation inseparable from their accomplishment.”
—George Gordon Noel Byron (1788–1824)
“Once a child has demonstrated his capacity for independent functioning in any area, his lapses into dependent behavior, even though temporary, make the mother feel that she is being taken advantage of....What only yesterday was a description of the child’s stage in life has become an indictment, a judgment.”
—Elaine Heffner (20th century)