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 .