category of commutative rings

Properties

Properties from the database

Deduced properties

Non-Properties

Non-Properties from the database

Deduced Non-Properties*

*This also uses the deduced properties.

Unknown properties

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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).