Difference between revisions of "Ideal"
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− | In [[ring theory]], an '''ideal''' is a special kind of [[subset]] of a [[ring]]. | + | In [[ring theory]], an '''ideal''' is a special kind of [[subset]] of a [[ring]]. Two-sided ideals in rings are the [[kernel]]s of ring [[homomorphism]]s; in this way, two-sided ideals of rings are similar to [[normal subgroup]]s of [[group]]s. |
Specifially, if <math>A</math> is a ring, a subset <math>\mathfrak{a}</math> of <math>A</math> is called a ''left ideal of <math>A</math>'' if it is a subgroup under addition, and if <math>xa \in \alpha</math>, for all <math>x\in R</math> and <math>a\in \mathfrak{a}</math>. Symbolically, this can be written as | Specifially, if <math>A</math> is a ring, a subset <math>\mathfrak{a}</math> of <math>A</math> is called a ''left ideal of <math>A</math>'' if it is a subgroup under addition, and if <math>xa \in \alpha</math>, for all <math>x\in R</math> and <math>a\in \mathfrak{a}</math>. Symbolically, this can be written as | ||
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== See also == | == See also == | ||
+ | * [[Quotient ring]] | ||
* [[Subring]] | * [[Subring]] | ||
− | |||
* [[Krull's Theorem]] | * [[Krull's Theorem]] | ||
* [[Pseudo-ring]] | * [[Pseudo-ring]] | ||
[[Category:Ring theory]] | [[Category:Ring theory]] |
Revision as of 14:16, 16 June 2008
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
.
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
.
Problems
<url>viewtopic.php?t=174516 Problem 1</url>