Let’s begin by verifying:

Theorem 1If are rings and is a homomorphism, then is an ideal of

*Proof:* Clearly so this set is nonempty. If then so Finally, if and then and so both and are in

In a sense, this is the only source of examples of ideals. This is shown by means of an abstract construction.

Theorem 2If is an ideal of a ring then there is a ring and a homomorphism such that

*Proof:* The proof resembles what we did to define the rings Begin by defining a relation on by ( iff ). Check that is an equivalence relation. We can then define where is the equivalence class of i.e.,

We turn into a ring by defining and Check that these operations are well defined. Then check that satisfies the axioms of rings.

Finally, let be the quotient map, Check that this is a homomorphism and that

Definition 3Anisomorphismis a bijective homomorphism. If is an isomorphism, we say that it is anautomorphism.

Proposition 4Suppose that is a field and is an ideal of Then either or

It will be very important for us to understand the automorphisms of the field extensions where For this, we will need some tools of linear algebra, so it will be useful to review Chapter 12 and Appendix C of the book.

*Typeset using LaTeX2WP. Here is a printable version of this post.*