**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.

Read more about this topic: Category Of Rings, Properties

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