Just as group homomorphisms preserve the group operation, ring and field homomorphisms preserve both operations.
Definition 4.2.1 (Ring homomorphism). A ring homomorphism is a map between rings that respects both operations and the identity:
for all ;
for all ;
.
A field homomorphism is a ring homomorphism between fields. A bijective ring homomorphism is an isomorphism.
Note. The first condition says that is a group homomorphism , so and automatically. The third condition does not follow from the second, even for fields: the zero map satisfies but . So we include it as a separate axiom. The inverse of an isomorphism is again a ring homomorphism: writing and , we get , and similarly for sums.
Example 4.2.2.
Reduction mod , the map sending an integer to its remainder on division by , is a ring homomorphism; this is the congruence property in the proof that is a ring.
Complex conjugation ,, is a field isomorphism, since , and .
The inclusion of a subring into a ring is a ring homomorphism.
Proposition 4.2.3 (Field homomorphisms are injective). If is a field homomorphism, then is injective.
Proof. Let with , and let . Then using in the field . If were , the left side would be ; hence . So the kernel of the additive group homomorphism is , and is injective (Proposition 2.5.6). ∎
Note. So every field homomorphism is an embedding: it identifies with the subfield of . Ring homomorphisms in general can collapse a great deal; reduction mod , for instance, sends all of to .