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Tame Algebra


Let A denote an R-algebra, so that A is a vector space over R and

 A×A->A
(1)
 (x,y)|->x·y,
(2)

where x·y is vector multiplication which is assumed to be bilinear. Now define

 Z={x in a:x·y=0 for some nonzero y in A},
(3)

where 0 in Z. A is said to be tame if Z is a finite union of subspaces of A. A two-dimensional 0-associative algebra is tame, but a four-dimensional 4-associative algebra and a three-dimensional 1-associative algebra need not be tame. It is conjectured that a three-dimensional 2-associative algebra is tame, and proven that a three-dimensional 3-associative algebra is tame if it possesses a multiplicative identity element.


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References

Finch, S. "Zero Divisor Structure in Real Algebras." http://algo.inria.fr/csolve/zerodiv/.

Referenced on Wolfram|Alpha

Tame Algebra

Cite this as:

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

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