Category of Rings - Properties - Morphisms

Morphisms

Unlike many categories studied in mathematics, there do not always exist morphisms between pairs of objects in Ring. This is a consequence of the fact that ring homomorphisms must preserve the identity. For example, there are no morphisms from the trivial ring 0 to any nontrivial ring. A necessary condition for there to be morphisms from R to S is that the characteristic of S divide that of R.

Note that even though some of the hom-sets are empty, the category Ring is still connected since it has an initial object.

Some special classes of morphisms in Ring include:

• Isomorphisms in Ring are the bijective ring homomorphisms.
• Monomorphisms in Ring are the injective homomorphisms. Not every monomorphism is regular however.
• Every surjective homomorphism is an epimorphism in Ring, but the converse is not true. The inclusion Z → Q is a nonsurjective epimorphism. The natural ring homomorphism from any commutative ring R to any one of its localizations is an epimorphism which is not necessarily surjective.
• The surjective homomorphisms can be characterized as the regular or extremal epimorphisms in Ring (these two classes coinciding).
• Bimorphisms in Ring are the injective epimorphisms. The inclusion Z → Q is an example of a bimorphism which is not an isomorphism.