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 next Augustan age will dawn on the other side of the Atlantic. There will, perhaps, be a Thucydides at Boston, a Xenophon at New York, and, in time, a Virgil at Mexico, and a Newton at Peru. At last, some curious traveller from Lima will visit England and give a description of the ruins of St Paul’s, like the editions of Balbec and Palmyra.”
—Horace Walpole (1717–1797)
“Do not require a description of the countries towards which you sail. The description does not describe them to you, and to- morrow you arrive there, and know them by inhabiting them.”
—Ralph Waldo Emerson (1803–1882)
“The next Augustan age will dawn on the other side of the Atlantic. There will, perhaps, be a Thucydides at Boston, a Xenophon at New York, and, in time, a Virgil at Mexico, and a Newton at Peru. At last, some curious traveller from Lima will visit England and give a description of the ruins of St. Paul’s, like the editions of Balbec and Palmyra.”
—Horace Walpole (1717–1797)