category of commutative rings

 algebra
Notation CRing\CRing Objects commutative rings Morphisms ring homomorphisms Parent CAlg(R)\CAlg(R) Related Ring\Ring, Rng\Rng, CMon\CMon External nLab link
This category is the special case of CAlg(R)\CAlg(R) where R=ZR = \IZ.

Satisfied properties

Assigned properties

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Properties from parent

Deduced properties

Unsatisfied properties

Assigned properties

Properties from parent

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

  • terminal object: zero ring
  • initial object: ring of integers
  • products: direct products with pointwise operations
  • coproducts: tensor products over Z\IZ

Special morphisms

  • isomorphisms: bijective morphisms
  • monomorphisms: injective morphisms
  • epimorphisms: A ring map f:R→Sf : R \to S is an epimorphism iff SS equals the dominion of f(R)⊆Sf(R) \subseteq S, meaning that for every s∈Ss \in S there is some matrix factorization (s)=YXZ(s) = Y X Z with X∈Mn×n(f(R))X \in M_{n \times n}(f(R)), Y∈M1×n(S)Y \in M_{1 \times n}(S), Z∈Mn×1(S)Z \in M_{n \times 1}(S) such that YX∈M1×n(f(R))YX \in M_{1 \times n}(f(R)) and XZ∈Mn×1(f(R))XZ \in M_{n \times 1}(f(R)).
  • regular monomorphisms:
  • regular epimorphisms: surjective morphisms

Indistinguishable categories

These categories in the database currently have exactly the same properties as the category of commutative rings. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.

Comments

  • Regular monomorphisms are discussed in MSE/695685, but probably they cannot be classified.

Functors

The database stores 1 functor based on the category of commutative rings.

Morphisms

The database stores 1 morphism based on the category of commutative rings.