It will be important to understand the subfields of a given field; this is a key step in figuring out whether a field is an extension by radicals or not. We need some “machinery” before we can develop this understanding.

Recall:

Definition 1Aringis a set together with two binary operations on such that:

- is commutative.
- There is an additive identity
- Any has an additive inverse
- is associative.
- is associative.
- distributes over both on the right and on the left.

So is a ring with the usual operations, as is each Any field is a ring. These are all examples of **commutative** rings, since holds in them. They are also rings **with identity**, since there is a multiplicative identity in all of them.

If is a field, is a commutative ring with identity.

Here is an exercise, to check your understanding: Suppose is a ring. Is a ring?

If denotes the collection of matrices with coefficients in the field then is a ring. In general, it is not commutative. Here is another exercise: Is there some such that is commutative?

If is a set and is its power set, i.e., the collection of all subsets of then we can turn into a commutative ring with identity: For subsets of define This is the **symmetric difference** of and Define Check that this is indeed a commutative ring with identity. What is the element? What is the element?

Definition 2If is a ring, asubringof is a subset which is a ring with the operations inherited from

Just as with subfields, there is an easy way of verifying whether a given subset of a ring is a subring.

Proposition 3Let be a ring and let Then is a subring of iff the following conditions hold:

- is closed under addition and multiplication.
- Whenever then

The proof of the proposition is an easy modification of the corresponding argument for subfields.

Here are some examples:

For any ring both and are subrings of

For any integer let be the set of multiples of Then is a subring of

In fact: Suppose is a subring. Then for some Indeed, either or else must contain a positive element. Using the division algorithm, one easily checks that if is the smallest positive element of then

Let An matrix has **eigenvector ** iff is a scalar multiple of i.e., for some complex number Fix a column vector with entries, and let be the collection of matrices in with eigenvector Then is a subring of Make sure you verify that this is indeed the case.

Definition 4Given rings and ahomomorphismbetween and is a function such that for any :

Informally, *translates* the operations of into the operations of in a “coherent” way.

Proposition 5If is a homomorphism of rings and then is a subring of

This is easy. It is a good idea to check first that for any

Here are some examples:

Let and and let be the map that to each assigns its class modulo Then is a homomorphism.

Let let and let be the evaluation by Then is a homomorphism. For example, if then This is not only a subring of it is in fact a field.

Proposition 6Suppose is a ring homomorphism. Then is 1-1 iff the only such that is in symbols,

For example, if and then is 1-1, because is transcendental, i.e., there is no nonzero polynomial in such that

We just saw that the image of a homomorphism is a subring of the target ring. Homomorphisms also provide us with subrings of the source ring. These are special subrings:

Definition 7Let be a ring. A subset is anidealprovided the following hold:

- Whenever then
- Whenever and then and

Lemma 8If is an ideal of a ring then is a subring of

Here is an example: Let let and let be the set of all polynomial multiples of Then is an ideal. It is the ideal **generated** by and we write

On the other hand, if is a column vector with 2 entries and is the collection of matrices in with eigenvector then this is a subring of but not necessarily an ideal, since there is no reason why should imply that is a multiple of for an arbitrary As an exercise, find a counterexample.

Proposition 9Let be a ring homomorphism. Let Then is an ideal of

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