# algebra without order

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

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: $\mathbb{R}^{2}$ with multiplication defined by $(x_{1},x_{2})*(y_{1},y_{2})=(x_{1}y_{1},0)$, ($(x_{1},x_{2}),(y_{1},y_{2})\in\mathbb{R}^{2}$) is not an algebra without order as multiplication of (0,1) with any other element gives (0,0).

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