Category of Rings - Properties - 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 ZQ 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 ZQ is an example of a bimorphism which is not an isomorphism.

Read more about this topic:  Category Of Rings, Properties

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