Abstract Algebra/Rings, ideals, ring homomorphisms

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Basic definitions[edit]

Definition 10.1:

A ring is a set together with two binary operations and and two special elements, the unit and the zero , such that:

  1. is an abelian group with respect to with neutral element .
  2. is a monoid (that is, a group without inversion) with respect to with neutral element .
  3. The distributive laws hold: , .

Examples 10.2:

  • The whole numbers with respect to usual addition and multiplication are a ring.
  • Every field is a ring.
  • If is a ring, then all polynomials over form a ring. This example will be explained later in the section on polynomial rings.

Definition 10.3:

Let be a ring. A left ideal of is a subset such that the following two things hold:

  1. is a subgroup of .
  2. , where (closedness by left multiplication).

Replacing closedness by left multiplication by closedness by right multiplication, we can define right ideals, and then both-sided ideals. If is a both-sided ideal of , we write .

We'll now show an important property of the set of all ideals of a given ring, namely that it's inductive. This means:

Definition 10.4:

Let be a partially ordered set (that is, the usual conditions transitivity, reflexivity and anti-symmetry are satisfied). is called inductive if and only if every ascending chain of elements of (that is, a sequence in such that ) has an upper bound (that is, an element such that ).

With this definition, we observe:

Theorem 10.5:

If a commutative ring is given, the set of all ideals of , partially ordered by inclusion (i.e. , where we use the convention of Donald Knuth and denote the power set of a set by ) is inductive.

Proof:

If

is an ascending chain of ideals, we set

and claim that . Indeed, if , find such that and . Then set , so that since . Similarly, if and , pick such that , whence since .

Residue class rings[edit]

Definition and theorem 10.4:

Let be a ring, and . Then we define a relation on as follows:

.

This relation is an equivalence relation, and an equivalence class shall be denoted by for . If we define an addition

and a multiplication

,

then these two are well-defined (i. e. independent of the choice of the representatives and ) and turn into a ring, called the residue class ring with respect to the ideal .

Proof:

First, we check that is an equivalence relation.

  1. Reflexiveness: since is an additive subgroup.
  2. Symmetry: since inverses are in the subgroup.
  3. Transitivity: Let and . Then , since a subgroup is closed under the group operation.

Then we check that addition and multiplication are well-defined. Let and . Then

for certain .

Furthermore,

for these same ; this is in by closedness by left and right multiplication.

The ring axioms directly carry over from the old ring .

Ring homomorphisms[edit]

Definition 10.5:

Let be rings. A ring homomorphism between the two is a map

such that:

  1. For all and .
  2. ( is the unit of and of ).