Associative Algebra

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

$$x\cdot(y\cdot z) = (x\cdot y)\cdot z,$$ (1)

where $\cdot$ denotes Multiplication. More formally, let $A$ denote an $\Bbb{R}$-algebra, so that $A$ is a Vector Space over $\Bbb{R}$ and

$$A\times A\to A$$ (2)

$$(x,y)\mapsto x\cdot y.$$ (3)

Then $A$ is said to be $m$-associative if there exists an $m$-D Subspace $S$ of $A$ such that

$$(y\cdot x)\cdot z=y\cdot(x\cdot z)$$ (4)

for all $y, z\in A$ and $x\in S$. Here, Vector Multiplication $x\cdot y$ is assumed to be Bilinear. An $n$-D $n$-associative Algebra is simply said to be ``associative.''

See also Associative

References

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