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:
“An intentional object is given by a word or a phrase which gives a description under which.”
—Gertrude Elizabeth Margaret Anscombe (b. 1919)
“It is possible—indeed possible even according to the old conception of logic—to give in advance a description of all ‘true’ logical propositions. Hence there can never be surprises in logic.”
—Ludwig Wittgenstein (1889–1951)
“I was here first introduced to Joe.... He was a good-looking Indian, twenty-four years old, apparently of unmixed blood, short and stout, with a broad face and reddish complexion, and eyes, methinks, narrower and more turned up at the outer corners than ours, answering to the description of his race. Besides his underclothing, he wore a red flannel shirt, woolen pants, and a black Kossuth hat, the ordinary dress of the lumberman, and, to a considerable extent, of the Penobscot Indian.”
—Henry David Thoreau (1817–1862)