Dunkl operator

2013-10-10T03:41:52Z

2.69.200.69:

In [[abstract algebra]], a '''nonassociative ring''' is a generalization of the concept of [[ring (mathematics)|ring]].

A nonassociative ring is a set ''R'' with two operations, addition and multiplication, such that:

# ''R'' is an [[abelian group]] under addition:
## <math>a+b = b+a</math>
## <math>(a+b)+c = a+(b+c)</math>
## There exists 0 in ''R'' such that <math>0 + a = a + 0 = a</math>
## For each ''a'' in ''R'', there exists an element −''a'' such that <math>a + (-a) = (-a) + a = 0</math>
# Multiplication is linear in each variable:
## <math>(a+b)c = ac + bc </math> (left distributive law)
## <math>a(b+c) = ab + ac </math> (right distributive law)

Unlike for rings, we do not require multiplication to satisfy [[associativity]]. We also do not require the presence of a unit, an element 1 such that <math>1x = x1 = x</math>.

In this context, ''nonassociative'' means that multiplication is not ''required'' to be associative, but associative multiplication is ''permitted''. Thus rings, which we'll call '''associative rings''' for clarity, are a special case of nonassociative rings.

Some classes of nonassociative rings replace associative laws with different constraints on the order of application of multiplication. For example [[Lie ring]]s and [[Lie algebra]]s replace the associative law with the [[Jacobi identity]], while [[Jordan ring]]s and [[Jordan algebra]]s replace the associative law with the [[Jordan identity]].

==Examples==
The [[octonion]]s, constructed by [[John T. Graves]] in 1843, were the first example of a ring that is not associative.
The [[hyperbolic quaternion]]s of [[Alexander Macfarlane]] (1891) form a nonassociative ring that suggested the mathematical footing for spacetime theory that followed later.

Other examples of nonassociative rings include the following:

* ('''R'''<sup>3</sup>, +, ×) where × is the [[cross product]] of vectors in 3-space
* The [[Cayley–Dickson construction]] provides an infinite family of nonassociative rings.
* [[Lie algebra]]s and [[Lie ring]]s
* [[Jordan algebra]]s and [[Jordan ring]]s
* [[Alternative ring]]s: A nonassociative ring R is said to be an alternative ring if [''x'',''x'',''y''] = [''y'',''x'',''x''] = 0, where [''x'',''y'',''z''] = (''xy'')''z'' − ''x''(''yz'') is the [[associator]].
* [[Semifield]]s (see [[quasifield]] axioms)

==Properties==
Most elementary properties of rings fail in the absence of associativity.
For example, for a nonassociative ring with an identity element:

* If an element <math>x</math> has left and right multiplicative inverses, <math>a^{L}</math> and <math>a^{R}</math>, then <math>a^{L}</math> and <math>a^{R}</math> can be distinct.
* Elements with multiplicative inverses can still be [[zero divisor]]s.

==References==
* Susumu Okubo (1995) ''Introduction to Octonian and Other Non-Associative Algebras in Physics'', [[Cambridge University Press]].

{{DEFAULTSORT:Nonassociative Ring}}
[[Category:Non-associative algebra]]

[[et:Ring (algebra)]]
[[es:Álgebra no asociativa]]

formulasearchengine - User contributions [en]

Dunkl operator