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:
“Science is intimately integrated with the whole social structure and cultural tradition. They mutually support one other—only in certain types of society can science flourish, and conversely without a continuous and healthy development and application of science such a society cannot function properly.”
—Talcott Parsons (1902–1979)
“As for types like my own, obscurely motivated by the conviction that our existence was worthless if we didn’t make a turning point of it, we were assigned to the humanities, to poetry, philosophy, painting—the nursery games of humankind, which had to be left behind when the age of science began. The humanities would be called upon to choose a wallpaper for the crypt, as the end drew near.”
—Saul Bellow (b. 1915)
“When I received this [coronation] ring I solemnly bound myself in marriage to the realm; and it will be quite sufficient for the memorial of my name and for my glory, if, when I die, an inscription be engraved on a marble tomb, saying, “Here lieth Elizabeth, which reigned a virgin, and died a virgin.””
—Elizabeth I (1533–1603)