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 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

Other articles related to "morphisms, morphism":

Composition Of Relations - Definition
... relations are sometimes regarded as the morphisms in a category Rel which has the sets as objects ... In Rel, composition of morphisms is exactly composition of relations as defined above ... sets is a subcategory of Rel that has the same objects but fewer morphisms ...
Nisnevich Topology - Definition
... A morphism of schemes f Y → X is called a Nisnevich morphism if it is an étale morphism such that for every (possibly non-closed) point x ∈ X, there exists a point y ∈ Y such ... A family of morphisms {uα Xα → X} is a Nisnevich cover if each morphism in the family is étale and for every (possibly non-closed) point x ∈ X, there exists α and a point y ∈ Xα s.t ... If the family is finite, this is equivalent to the morphism from to X being a Nisnevich morphism ...
Mathematical Object - Category Theory
... which abstracts sets as objects and the operations thereon as morphisms between those objects ... vertices of a graph whose edges as the morphisms abstract the ways in which those objects can transform and whose structure is encoded in the composition law for morphisms ...
Factorization System
... A factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that E and M both contain all isomorphisms of C and are closed under ... Every morphism f of C can be factored as for some morphisms and ... The factorization is functorial if and are two morphisms such that for some morphisms and, then there exists a unique morphism making the following diagram commute ...
List Of Zero Terms - Zero Morphisms
... A zero morphism in a category is a generalised absorbing element under function composition any morphism composed with a zero morphism gives a zero morphism ... Specifically, if 0XY X → Y is the zero morphism among morphisms from X to Y, and f A → X and g Y → B are arbitrary morphisms, then g ∘ 0XY = 0XB and 0XY ∘ f ... If a category has a zero object 0, then there are canonical morphisms X → 0 and 0 → Y, and composing them gives a zero morphism 0XY X → Y ...