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Difference between revisions of "Lie ring"

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A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058740/l0587401.png" /> that satisfies the conditions
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A ring $A$ that satisfies the conditions
  
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<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/l/l058/l058740/l0587402.png" /></td> </tr></table>
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$$a^2=0$$
  
 
and
 
and
  
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<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/l/l058/l058740/l0587403.png" /></td> </tr></table>
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$$(ab)c+(bc)a+(ca)b=0$$
  
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(the Jacobi identity), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058740/l0587404.png" /> are any elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058740/l0587405.png" />. The first of these conditions implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058740/l0587406.png" /> is anti-commutative:
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(the Jacobi identity), where $a,b,c$ are any elements of $A$. The first of these conditions implies that $A$ is anti-commutative:
  
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<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/l/l058/l058740/l0587407.png" /></td> </tr></table>
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$$ba=-ab.$$
  
 
The Lie rings form a [[Variety of rings|variety of rings]], in general non-associative. It contains, however, all rings with zero multiplication.
 
The Lie rings form a [[Variety of rings|variety of rings]], in general non-associative. It contains, however, all rings with zero multiplication.
  
 
See also [[Non-associative rings and algebras|Non-associative rings and algebras]].
 
See also [[Non-associative rings and algebras|Non-associative rings and algebras]].

Latest revision as of 16:01, 22 July 2014

A ring $A$ that satisfies the conditions

$$a^2=0$$

and

$$(ab)c+(bc)a+(ca)b=0$$

(the Jacobi identity), where $a,b,c$ are any elements of $A$. The first of these conditions implies that $A$ is anti-commutative:

$$ba=-ab.$$

The Lie rings form a variety of rings, in general non-associative. It contains, however, all rings with zero multiplication.

See also Non-associative rings and algebras.

How to Cite This Entry:
Lie ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_ring&oldid=15526
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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