Difference between revisions of "Lie ring"
From Encyclopedia of Mathematics
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− | A ring | + | {{TEX|done}} |
+ | A ring $A$ that satisfies the conditions | ||
− | + | $$a^2=0$$ | |
and | and | ||
− | + | $$(ab)c+(bc)a+(ca)b=0$$ | |
− | (the Jacobi identity), where | + | (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|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=32508
Lie ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_ring&oldid=32508
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article