Ideal
In ring theory, an ideal is a special kind of subset of a ring. Two-sided ideals in rings are the kernels of ring homomorphisms; in this way, two-sided ideals of rings are similar to normal subgroups of groups.
Specifially, if is a ring, a subset
of
is called a left ideal of
if it is a subgroup under addition, and if
, for all
and
. Symbolically, this can be written as
A right ideal is defined similarly, but with the modification
. If
is both a left ideal and a right ideal, it is called a two-sided ideal. In a commutative ring, all three ideals are the same; they are simply called ideals. Note that the right ideals of a ring
are exactly the left ideals of the opposite ring
.
An ideal has the structure of a pseudo-ring, that is, a structure that satisfies the properties of rings, except possibly for the existance of a multiplicative identity.
By abuse of language, a (left, right, two-sided) ideal of a ring is called maximal if it is a maximal element of the set of (left, right, two-sided) ideals distinct from
.
Examples of Ideals
In the ring , the ideals are the rings of the form
, for some integer
.
In a field , the only ideals are the set
and
itself.
In general, if is a ring and
is an element of
, the set
is a left ideal of
. Ideals of this form are known as principle ideals.
Generated Ideals
Let be a ring, and let
be a family of elements of
. The left ideal generated by the family
is the set of elements of
of the form
where
is a family of elements of
of finite support, as this set is a left ideal of
, thanks to distributivity, and every element of the set must be in every left ideal containing
. Similarly, the two-sided ideal generated by
is the set of elements of
of the form
where
and
are families of finite support.
If is a set of (left, right, two-sided) ideals of
, then the (left, two sided) ideal generated by
is the set of elements of the form
, where
is an element of
and
is a family of finite support. For this reason, the ideal generated by the
is sometimes denoted
.
Multiplication of Ideals
If and
are two-sided ideals of a ring
, then the set of elements of the form
, for
and
, is also an ideal of
. It is called the product of
and
, and it is denoted
. It is generated by the elements of the form
, for
and
. Since
and
are two-sided ideals,
is a subset of both
and of
, so
The ideals of
constitute a pseudo-ring.
Proposition 1. Let and
be two-sided ideals of a ring
such that
, for each index
. Then
Proof. We induct on . For
, the proposition is degenerately true.
Now, suppose the proposition holds for . Then
which proves the proposition.
Problems
<url>viewtopic.php?t=174516 Problem 1</url>