Commutative Algebra

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

$$A\times A\to A$$

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

Now define

$$Z\equiv\{x\in A: x\cdot y=0 {\rm\ for\ some\ } y\in A\not=0\},$$

where $0\in Z$. An Associative $\Bbb{R}$-algebra is commutative if $x\cdot y=y\cdot x$ for all $x, y\in A$. Similarly, a Ring is commutative if the Multiplication operation is commutative, and a Lie Algebra is commutative if the Commutator $[A,B]$ is 0 for every $A$ and $B$ in the Lie Algebra.

See also Abelian Group, Commutative

References

Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and Algorithms: An Introduction to Algebraic Geometry and Commutative Algebra, 2nd ed. New York: Springer-Verlag, 1996.

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

MacDonald, I. G. and Atiyah, M. F. Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1969.

Samuel, P. and Zariski, O. Commutative Algebra, Vol. 2. New York: Springer-Verlag, 1997.