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Associative Algebra


In simple terms, let x, y, and z be members of an algebra. Then the algebra is said to be associative if

 x·(y·z)=(x·y)·z,
(1)

where · denotes multiplication. More formally, let A denote an R-algebra, so that A is a vector space over R and

 A×A->A
(2)
 (x,y)|->x·y.
(3)

Then A is said to be m-associative if there exists an m-dimensional subspace S of A such that

 (y·x)·z=y·(x·z)
(4)

for all y,z in A and x in S. Here, vector multiplication x·y is assumed to be bilinear. An n-dimensional n-associative algebra is simply said to be "associative."


See also

Associative

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References

Finch, S. "Zero Divisor Structure in Real Algebras." http://algo.inria.fr/csolve/zerodiv/.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.

Referenced on Wolfram|Alpha

Associative Algebra

Cite this as:

Weisstein, Eric W. "Associative Algebra." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AssociativeAlgebra.html

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