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
Other articles related to "morphisms, morphism":
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... 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 ...
... 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 ...
... 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 ...
... 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 ...
... 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 ...
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