Definition:Zero Divisor/Algebra
Definition
Let $\struct {A_R, \oplus}$ be an algebra over a ring $\struct {R, +, \cdot}$.
Let the zero vector of $A_R$ be $\mathbf 0_R$.
Let $a, b \in A_R$ such that $b \ne \mathbf 0_R$.
Then $a$ is a zero divisor of $A_R$ if and only if:
- $a \oplus b = \mathbf 0_R$
Also defined as
Some sources define a zero divisor as an element $x \in R_{\ne 0_R}$ such that:
- $\exists y \in R_{\ne 0_R}: x \circ y = 0_R$
where $R_{\ne 0_R}$ is defined as $R \setminus \set {0_R}$.
That is, the element $0_R$ itself is not classified as a zero divisor.
This definition is the same as the one given on this website as a proper zero divisor.
Also known as
Some sources hyphenate, as: zero-divisor.
Some sources run the words together: zerodivisor.
Some use the more explicit and pedantic divisor of zero.
Warning
Beware the terminology divisor of zero.
It is easy to confuse this with the fact that, for every element $a$ of a ring $R$, from Ring Product with Zero we have that $a \circ 0_R = 0_R$.
Hence one may say that every such element is a divisor of zero.
However, the concept of a zero divisor specifically requires that the $b$ in $a \circ b = 0_R$ is not zero.
Notation
The conventional notation for $x$ is a divisor of $y$ is "$x \mid y$", but there is a growing trend to follow the notation "$x \divides y$", as espoused by Knuth etc.
From Ronald L. Graham, Donald E. Knuth and Oren Patashnik: Concrete Mathematics: A Foundation for Computer Science (2nd ed.):
- The notation '$m \mid n$' is actually much more common than '$m \divides n$' in current mathematics literature. But vertical lines are overused -- for absolute values, set delimiters, conditional probabilities, etc. -- and backward slashes are underused. Moreover, '$m \divides n$' gives an impression that $m$ is the denominator of an implied ratio. So we shall boldly let our divisibility symbol lean leftward.
An unfortunate unwelcome side-effect of this notational convention is that to indicate non-divisibility, the conventional technique of implementing $/$ through the notation looks awkward with $\divides$, so $\not \! \backslash$ is eschewed in favour of $\nmid$.
Some sources use $\ \vert \mkern -10mu {\raise 3pt -} \ $ or similar to denote non-divisibility.
Also see
- Results about zero divisors can be found here.