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Encyclopedia > Nonassociative ring

In abstract algebra, a nonassociative ring is a generalization of the concept of ring. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...


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

  1. R is an abelian group under addition:
    1. a + b = b + a
    2. (a + b) + c = a + (b + c)
    3. There exists 0 in R such that 0 + a = a + 0 = a
    4. For each a in R, there exists an element -a such that a + ( − a) = ( − a) + a = 0
  2. Multiplication is linear in each variable:
    1. (a + b)c = ac + bc (left distributive law)
    2. a(b + c) = ab + ac (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 1x = x1 = x. In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In mathematics, associativity is a property that a binary operation can have. ...


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 rings and Lie algebras replace the associative law with the Jacobi identity, while Jordan rings and Jordan algebras replace the associative law with the Jordan identity. In mathematics a Lie ring is a structure related to Lie algebras that can arise as a generalisation of Lie algebras, or through the study of the Lower central series of groups. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. ... In mathematics, a Jordan algebra is defined in abstract algebra as a (usually nonassociative) algebra over a field with multiplication satisfying the following axioms: (commutative law) (Jordan identity) Jordan algebras were first introduced by Pascual Jordan in quantum mechanics. ... In mathematics, a Jordan algebra is defined in abstract algebra as an algebra over a field with multiplication satisfying the following axioms: (commutative law) (Jordan identity) Jordan algebras were first introduced by Pascual Jordan in quantum mechanics. ... In mathematics, a Jordan algebra is defined in abstract algebra as a (usually nonassociative) algebra over a field with multiplication satisfying the following axioms: (commutative law) (Jordan identity) Jordan algebras were first introduced by Pascual Jordan in quantum mechanics. ...


Examples

The octonions, constructed by John T. Graves in 1843, were the first example of a ring that is not associative. The hyperbolic quaternions of Alexander MacFarlane form a nonassociative ring that suggested the mathematical footing for spacetime theory that followed later. In mathematics, the octonions are a nonassociative extension of the quaternions. ... Year 1843 (MDCCCXLIII) was a common year starting on Sunday (link will display the full calendar) of the Gregorian Calendar (or a common year starting on Friday of the 12-day slower Julian calendar). ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Alexander Macfarlane (Blairgowrie, Scotland, April 21, 1851 – Chatham, Ontario, August 28, 1913) was a Scottish-Canadian logician, physicist, and mathematician. ...


Other examples of nonassociative rings include the following:

  • The Cayley-Dickson construction provides an infinite family of nonassociative rings.
  • Lie algebras and Lie rings
  • Jordan algebras and Jordan rings.
  • Alternative rings: 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.

In mathematics, the Cayley-Dickson construction produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics a Lie ring is a structure related to Lie algebras that can arise as a generalisation of Lie algebras, or through the study of the Lower central series of groups. ... In mathematics, a Jordan algebra is defined in abstract algebra as an algebra over a field with multiplication satisfying the following axioms: (commutative law) (Jordan identity) Jordan algebras were first introduced by Pascual Jordan in quantum mechanics. ... In mathematics, a Jordan algebra is defined in abstract algebra as a (usually nonassociative) algebra over a field with multiplication satisfying the following axioms: (commutative law) (Jordan identity) Jordan algebras were first introduced by Pascual Jordan in quantum mechanics. ... In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative only alternative. ... For a nonassociative ring , the associator is the map defined by the expression The associator is linear in each variable. ...

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 x has left and right inverses, aL and aR, then aL and aR can be distinct.
  • Elements with multiplicative inverses can still be zero divisors.

  Results from FactBites:
 
Ring (mathematics) - Wikipedia, the free encyclopedia (1393 words)
A ring is a generalization of the integers, which itself is an example of a ring.
Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings.
For n>2, this matrix ring is an example of a noncommutative ring (unless R is the trivial ring).
  More results at FactBites »

 
 

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