Zero Divisor

A Nonzero element $x$ of a Ring for which $x\cdot y=0$, where $y$ is some other Nonzero element and the vector multiplication $x\cdot y$ is assumed to be Bilinear. A Ring with no zero divisors is known as an Integral Domain. Let $A$ denote an $\Bbb{R}$-algebra, so that $A$ is a Vector Space over $R$ and

$$A\times A\to A$$

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

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 $m$-Associative if there exists an $m$-dimensional Subspace $S$ of $A$ such that $(y\cdot x)\cdot z=y\cdot(x\cdot z)$ for all $y, z\in A$ and $x\in S$. $A$ is said to be Tame if $Z$ is a finite union of Subspaces of $A$.

References

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