category of commutative rings

Satisfied Properties

Properties from the database

Deduced properties

Unsatisfied Properties

Properties from the database

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\mathbb{Z}

Special morphisms

  • isomorphisms: bijective ring homomorphisms
  • monomorphisms: injective ring homomorphisms
  • 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(R)X \in M_{n \times n}(R), Y∈M1×n(S)Y \in M_{1 \times n}(S), and Z∈Mn×1(S)Z \in M_{n \times 1}(S).
  • regular monomorphisms:
  • regular epimorphisms: surjective homomorphisms

Undistinguishable 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.