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; and are identical, but since f is a monomorphism this is impossible. However, surjective ring homomorphisms are vastly different from epimorphisms in the category of rings. For example, the inclusion is a ring epimorphism, but not a surjection. However, they are exactly the same as the strong epimorphisms. See also Change of rings Ring extension Citations Notes References Ring theory Morphisms