Ring homomorphisms and Factor rings

Let be rings. Then is a ring homomorphism if

1. is a group homomorphism with respect to , meaning and
2. is a monoid homomorphism with respect to , meaning

Let be a ring homomorphism.

1. .
2. is a subring of .
3. is a group homomorphism with respect to
4. is a subring of .

Let be a ring homomorphism, and a field. Then is injective.

Proof:
, so either or . But , so .

Let and be ring homomorphisms. Then is a ring homomorphism.

Also, if is bijective, then is a ring homomorphism.

Let be rings. The following are ring homomorphisms.

1. The inclusion map .
2. We can send formal polynomial to polynomial functions by
   
   . Note that in general, is not injective.
   For example, let . Then , but .
3. For a fixed , .

Let be a ring and . Then

1. defines an equivalence relation on . The equivalence classes .
2. The set of all equivalence classes is a ring with This is called a factor ring or quotient ring, or residue class ring.
3. The canonical projection is a surjectve ring homomorphism with .

Proof:
Recall that is an abelian group. This means is a normal subgroup. This means is a factor group and that is a surjective group homomorphism, and .

We still need to show that the multiplication is well defined. Let and . We know and . This means since every term has a factor in and .

Hence, is also a monoid homomorphism with respect to .

Let be a ring homomorphism and . Then there is a unique ring homomorphism such that .

4. is injective .

Proof:
We already know that there is a unique group homomorphism with the same properties. Left to show is that is a ring homomorphism. Exercise!

Let be a surjective ring homomorphism and . Then .

Constructive way:
Can also use the universal property.

Let be a ring and a subring, and . Then .

Let be a ring and be ideals. Then .