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''idempotent element''
 
''idempotent element''
  
An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i0500801.png" /> of a ring, semi-group or groupoid equal to its own square: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i0500802.png" />. An idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i0500803.png" /> is said to contain an idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i0500804.png" /> (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i0500805.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i0500806.png" />. For associative rings and semi-groups, the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i0500807.png" /> is a partial order on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i0500808.png" /> of idempotent elements, called the natural partial order on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i0500809.png" />. Two idempotents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i05008010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i05008011.png" /> of a ring are said to be orthogonal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i05008012.png" />. With every idempotent of a ring (and also with every system of orthogonal idempotents) there is associated the so-called [[Peirce decomposition|Peirce decomposition]] of the ring. For an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i05008013.png" />-ary algebraic relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i05008014.png" />, an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i05008015.png" /> is said to be an idempotent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i05008016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i05008017.png" /> occurs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i05008018.png" /> times between the brackets.
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An element $e$ of a ring, semi-group or groupoid equal to its own square: $e^2=e$. An idempotent $e$ is said to contain an idempotent $f$ (denoted by $e\geq f$) if $ef=e=fe$. For associative rings and semi-groups, the relation $\geq$ is a partial order on the set $E$ of idempotent elements, called the natural partial order on $E$. Two idempotents $u$ and $v$ of a ring are said to be orthogonal if $uv=0=vu$. With every idempotent of a ring (and also with every system of orthogonal idempotents) there is associated the so-called [[Peirce decomposition|Peirce decomposition]] of the ring. For an $n$-ary algebraic relation $\omega$, an element $e$ is said to be an idempotent if $(e\ldots e)\omega=e$, where $e$ occurs $n$ times between the brackets.
  
  
  
 
====Comments====
 
====Comments====
An algebraic operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i05008019.png" /> is sometimes said to be idempotent if every element of the set on which it acts is idempotent in the sense defined above. Such operations are also called affine operations; the latter name is preferable because an affine unary operation is not the same thing as an idempotent element of the semi-group of unary operations. In the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i05008020.png" />-modules, the affine operations are those of the form
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An algebraic operation $\omega$ is sometimes said to be idempotent if every element of the set on which it acts is idempotent in the sense defined above. Such operations are also called affine operations; the latter name is preferable because an affine unary operation is not the same thing as an idempotent element of the semi-group of unary operations. In the theory of $R$-modules, the affine operations are those of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i05008021.png" /></td> </tr></table>
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$$(x_1,\ldots,x_n)\mapsto\sum_{i=1}^nr_ix_i$$
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050080/i05008022.png" />.
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with $\sum_{i=1}^nr_i=1$.

Revision as of 15:18, 1 August 2014

idempotent element

An element $e$ of a ring, semi-group or groupoid equal to its own square: $e^2=e$. An idempotent $e$ is said to contain an idempotent $f$ (denoted by $e\geq f$) if $ef=e=fe$. For associative rings and semi-groups, the relation $\geq$ is a partial order on the set $E$ of idempotent elements, called the natural partial order on $E$. Two idempotents $u$ and $v$ of a ring are said to be orthogonal if $uv=0=vu$. With every idempotent of a ring (and also with every system of orthogonal idempotents) there is associated the so-called Peirce decomposition of the ring. For an $n$-ary algebraic relation $\omega$, an element $e$ is said to be an idempotent if $(e\ldots e)\omega=e$, where $e$ occurs $n$ times between the brackets.


Comments

An algebraic operation $\omega$ is sometimes said to be idempotent if every element of the set on which it acts is idempotent in the sense defined above. Such operations are also called affine operations; the latter name is preferable because an affine unary operation is not the same thing as an idempotent element of the semi-group of unary operations. In the theory of $R$-modules, the affine operations are those of the form

$$(x_1,\ldots,x_n)\mapsto\sum_{i=1}^nr_ix_i$$

with $\sum_{i=1}^nr_i=1$.

How to Cite This Entry:
Idempotent. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Idempotent&oldid=13382
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article