Definition:Zero Divisor/Ring

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Definition

Let $\struct {R, +, \circ}$ be a ring.


A zero divisor (in $R$) is an element $x \in R$ such that either:

$\exists y \in R^*: x \circ y = 0_R$

or:

$\exists y \in R^*: y \circ x = 0_R$

where $R^*$ is defined as $R \setminus \set {0_R}$.


That is, such that $x$ is either a left zero divisor or a right zero divisor.


The expression:

$x$ is a zero divisor

can be written:

$x \divides 0_R$


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 known as

Some sources hyphenate, as: zero-divisor. Some run the words together: zerodivisor.

Some use the more explicit and pedantic divisor of zero.


Also defined as

Some sources define a zero-divisor as an element $x \in R^*$ such that:

$\exists y \in R^*: x \circ y = 0_R$

or:

$\exists y \in R^*: y \circ x = 0_R$

where $R^*$ 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 Proper Zero Divisor.


Also see

  • Results about zero divisors can be found here.


Sources