algebra without order

An algebra (http://planetmath.org/Algebra) $A$ is said to be order without order if it is commutative and
for each $a\in A$, there exists $b\in A$ such that $ab\ne 0$.

The phrase algebra without order seems first in the book “Multipliers of Banach algebras” by Ronald Larsen. In noncommutative case, the concept is divied into two parts – without left/right order. However, in the noncommutative case, it is defined in terms of the injectivity of the left (right) regular representation given by $x\in A\mapsto L_x\in L(A)$.

Note that for an algebra $A$ and an element $x\in A$, $L_x:A\to A$ is the map defined by $L_x(y)=xy$. Then $L_x$ is a linear operator on $A$. It is easy to see that $A$ is without left order if and only if the map $x\in A\mapsto L_x\in L(A)$ is one-one; equivalently, the left ideal $\{x\in A:x\in A\}=\{0\}$. This ideal is is called the left annihilator of $A$.

Every commutative algebra with identity is without order.

Example: ${ℝ}^2$ with multiplication defined by $(x_1,x_2)*(y_1,y_2)=(x_1{y}_1,0)$, ($(x_1,x_2),(y_1,y_2)\in {ℝ}^2$) is not an algebra without order as multiplication of (0,1) with any other element gives (0,0).

Title	algebra without order
Canonical name	AlgebraWithoutOrder
Date of creation	2013-03-22 14:46:01
Last modified on	2013-03-22 14:46:01
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	11
Author	mathcam (2727)
Entry type	Definition
Classification	msc 13A99