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)
“If there is nothing new on the earth, still the traveler always has a resource in the skies. They are constantly turning a new page to view. The wind sets the types on this blue ground, and the inquiring may always read a new truth there.”
—Henry David Thoreau (1817–1862)
“Look how my ring encompasseth thy finger;
Even so thy breast encloseth my poor heart.
Wear both of them, for both of them are thine.”
—William Shakespeare (1564–1616)