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Difference between revisions of "Ideal"

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<cmath> \sum_{i\in I} a_i x_i b_i, </cmath>
 
<cmath> \sum_{i\in I} a_i x_i b_i, </cmath>
 
where <math>(a_i)_{i\in I}</math> and <math>(b_i)_{i \in I}</math> are families of finite support.
 
where <math>(a_i)_{i\in I}</math> and <math>(b_i)_{i \in I}</math> are families of finite support.
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If <math>(\mathfrak{a}_i)_{i\in I}</math> is a set of (left, right, two-sided) ideals of <math>A</math>, then the (left, two sided) ideal generated by <math>\bigcup_{i\in I} \mathfrak{a}_i</math> is the set of elements of the form <math>\sum_i x_i</math>, where <math>x_i</math> is an element of <math>\mathfrak{a}_i</math> and <math>(x_i)_{i\in I}</math> is a family of finite support.  For this reason, the ideal generated by the <math>\mathfrak{a}_i</math> is sometimes denoted <math>\sum_{i\in I} \mathfrak{a}_i</math>.
  
 
==Problems==
 
==Problems==

Revision as of 15:11, 13 June 2008

In ring theory, an ideal is a special kind of subset of a ring.

Specifially, if $A$ is a ring, a subset $\mathfrak{a}$ of $A$ is called a left ideal of $A$ if it is a subgroup under addition, and if $xa \in \alpha$, for all $x\in R$ and $a\in \mathfrak{a}$. Symbolically, this can be written as \[0\in \mathfrak{a}, \qquad \mathfrak{a+a\subseteq a}, \qquad A \mathfrak{a \subseteq a} .\] A right ideal is defined similarly, but with the modification $\mathfrak{a}A \subseteq \mathfrak{a}$. If $\mathfrak{a}$ 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 $A$ are exactly the left ideals of the opposite ring $A^0$.

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 $A$ is called maximal if it is a maximal element of the set of (left, right, two-sided) ideals distinct from $A$.

Examples of Ideals

In the ring $\mathbb{Z}$, the ideals are the rings of the form $n \mathbb{Z}$, for some integer $n$.

In a field $F$, the only ideals are the set $\{0\}$ and $F$ itself.

In general, if $A$ is a ring and $x$ is an element of $A$, the set $Ax$ is a left ideal of $A$.

Generated Ideals

Let $A$ be a ring, and let $(x_i)_{i\in I}$ be a family of elements of $A$. The left ideal generated by the family $(x_i)_{i\in I}$ is the set of elements of $A$ of the form \[\sum_{i \in I} a_i x_i,\] where $(a_i)_{i \in I}$ is a family of elements of $A$ of finite support, as this set is a left ideal of $A$, thanks to distributivity, and every element of the set must be in every left ideal containing $(x_i)_{i\in I}$. Similarly, the two-sided ideal generated by $(x_i)_{i\in I}$ is the set of elements of $A$ of the form \[\sum_{i\in I} a_i x_i b_i,\] where $(a_i)_{i\in I}$ and $(b_i)_{i \in I}$ are families of finite support.

If $(\mathfrak{a}_i)_{i\in I}$ is a set of (left, right, two-sided) ideals of $A$, then the (left, two sided) ideal generated by $\bigcup_{i\in I} \mathfrak{a}_i$ is the set of elements of the form $\sum_i x_i$, where $x_i$ is an element of $\mathfrak{a}_i$ and $(x_i)_{i\in I}$ is a family of finite support. For this reason, the ideal generated by the $\mathfrak{a}_i$ is sometimes denoted $\sum_{i\in I} \mathfrak{a}_i$.

Problems

<url>viewtopic.php?t=174516 Problem 1</url>

See also

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