Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra In mathematics, a ring homomorphism is a structure-preserving function between two rings.
If in addition f is a bijection, then its inverse f−1 is also a ring homomorphism.
If R and S are rngs, then the corresponding notion is that of a rng homomorphism,[a] defined as above except without the third condition f(1R) = 1S.
Then, directly from these definitions, one can deduce: Moreover, Injective ring homomorphisms are identical to monomorphisms in the category of rings: If f : R → S is a monomorphism that is not injective, then it sends some r1 and r2 to the same element of S. Consider the two maps g1 and g2 from Z[x] to R that map x to r1 and r2, respectively; f ∘ g1 and f ∘ g2 are identical, but since f is a monomorphism this is impossible.
For example, the inclusion Z ⊆ Q is a ring epimorphism, but not a surjection.