A ring is a set together with two binary operations and and two special elements, the unit and the zero , such that:
- is an abelian group with respect to with neutral element .
- is a monoid (that is, a group without inversion) with respect to with neutral element .
- The distributive laws hold: , .
- 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.
Let be a ring. A left ideal of is a subset such that the following two things hold:
- is a subgroup of .
- , 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:
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:
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.
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
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 .
First, we check that is an equivalence relation.
- Reflexiveness: since is an additive subgroup.
- Symmetry: since inverses are in the subgroup.
- 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 .
for these same ; this is in by closedness by left and right multiplication.
The ring axioms directly carry over from the old ring .
Let be rings. A ring homomorphism between the two is a map
- For all and .
- ( is the unit of and of ).