Types of Ring Homomorphisms
A bijective ring homomorphism is called a ring isomorphism. A ring homomorphism whose domain is the same as its range is called a ring endomorphism. A ring automorphism is a bijective endomorphism.
Injective ring homomorphisms are identical to monomorphisms in the category of rings: If f:R→S is a monomorphism which is not injective, then it sends some r1 and r2 to the same element of S. Consider the two maps g1 and g2 from Z to R which map x to r1 and r2, respectively; f g1 and f g2 are identical, but since f is a monomorphism this is impossible.
However, surjective ring homomorphisms are vastly different from epimorphisms in the category of rings. For example, the inclusion Z ⊆ Q is a ring epimorphism, but not a surjection. However, they are exactly the same as the strong epimorphisms.
Read more about this topic: Ring Homomorphism
Famous quotes containing the words types of, types and/or ring:
“The wider the range of possibilities we offer children, the more intense will be their motivations and the richer their experiences. We must widen the range of topics and goals, the types of situations we offer and their degree of structure, the kinds and combinations of resources and materials, and the possible interactions with things, peers, and adults.”
—Loris Malaguzzi (19201994)
“Our major universities are now stuck with an army of pedestrian, toadying careerists, Fifties types who wave around Sixties banners to conceal their record of ruthless, beaverlike tunneling to the top.”
—Camille Paglia (b. 1947)
“The life of man is a self-evolving circle, which, from a ring imperceptibly small, rushes on all sides outwards to new and larger circles, and that without end.”
—Ralph Waldo Emerson (18031882)