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Difference between revisions of "Ideal"

(New page: In ring theory, an ideal is a special subset of the ring. ==Definition== Let <math>R</math> be a ring, with <math>(R, +)</math> the underlying additive group of the ring. A subset <m...)
 
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In [[ring]] theory, an ideal is a special subset of the ring.  
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In [[ring theory]], an '''ideal''' is a special kind of [[subset]] of a [[ring]].  
  
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==Definition==
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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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Let <math>R</math> be a ring, with <math>(R, +)</math> the underlying additive group of the ring. A subset <math>I</math> of <math>R</math> is called right ideal of <math>R</math> if
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<cmath> 0\in \mathfrak{a}, \qquad \mathfrak{a+a\subseteq a}, \qquad A \mathfrak{a \subseteq a} . </cmath>
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- <math>(I, +)</math> is a subgroup of <math>(R, +)</math>
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A ''right ideal'' is defined similarly, but with the modification <math>\mathfrak{a}A \subseteq \mathfrak{a}</math>.  If <math>\mathfrak{a}</math> 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 <math>A</math> are exactly the left ideals of the opposite ring <math>A^0</math>.
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- <math>xr</math> is in <math>I</math> for all <math>x</math> in <math>I</math> and all <math>r</math> in <math>R</math>
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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.
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By abuse of language, a (left, right, two-sided) ideal of a ring <math>A</math> is called ''maximal'' if it is a [[maximum |maximal element]] of the set of (left, right, two-sided) ideals distinct from <math>A</math>.
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== Examples of Ideals ==
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In the ring <math>\mathbb{Z}</math>, the ideals are the rings of the form <math>n \mathbb{Z}</math>, for some integer <math>n</math>.
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In a [[field]] <math>F</math>, the only ideals are the set <math>\{0\}</math> and <math>F</math> itself.
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In general, if <math>A</math> is a ring and <math>x</math> is an element of <math>A</math>, the set <math>Ax</math> is a left ideal of <math>A</math>.
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== Generated Ideals ==
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Let <math>A</math> be a ring, and let <math>(x_i)_{i\in I}</math> be a family of elements of <math>A</math>.  The left ideal generated by the family <math>(x_i)_{i\in I}</math> is the set of elements of <math>A</math> of the form
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<cmath> \sum_{i \in I} a_i x_i, </cmath>
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where <math>(a_i)_{i \in I}</math> is a family of elements of <math>A</math> of [[finite]] [[support]], as this set is a left ideal of <math>A</math>, thanks to distributivity, and every element of the set must be in every left ideal containing <math>(x_i)_{i\in I}</math>.  Similarly, the two-sided ideal generated by <math>(x_i)_{i\in I}</math> is the set of elements of <math>A</math> of the form
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<cmath> \sum_{i\in I} a_i x_i b_i, </cmath>
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where <math>(a_i)_{i\in I}</math> and <math>(b_i)_{i \in I}</math> are families of finite support.
  
 
==Problems==
 
==Problems==
 
<url>viewtopic.php?t=174516 Problem 1</url>
 
<url>viewtopic.php?t=174516 Problem 1</url>
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== See also ==
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* [[Subring]]
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* [[Quotient ring]]
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* [[Krull's Theorem]]
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* [[Pseudo-ring]]
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[[Category:Ring theory]]

Revision as of 09:55, 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.

Problems

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

See also

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