Definition:Zero Divisor/Algebra

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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. GrahamDonald 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.