Tame Algebra

Let $A$ denote an $\Bbb{R}$-algebra, so that $A$ is a Vector Space over $R$ and

$$A\times A\to A$$

$$(x,y)\mapsto x\cdot y,$$

where $x\cdot y$ is vector multiplication which is assumed to be Bilinear. Now define

$$Z\equiv\{x\in a: x\cdot y=0{\rm\ for\ some\ nonzero\ } y\in A\},$$

where $0\in Z$. $A$ is said to be tame if $Z$ is a finite union of Subspaces of $A$. A 2-D 0-Associative algebra is tame, but a 4-D 4-Associative algebra and a 3-D 1-Associative algebra need not be tame. It is conjectured that a 3-D 2-Associative algebra is tame, and proven that a 3-D 3-Associative algebra is tame if it possesses a multiplicative Identity Element.

References

Finch, S. ``Zero Structures in Real Algebras.'' http://www.mathsoft.com/asolve/zerodiv/zerodiv.html.