Definition:Zero Divisor/Ring
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:
- $x \divides 0_R$
Left Zero Divisor
A left zero divisor (in $R$) is an element $x \in R$ such that:
- $\exists y \in R^*: x \circ y = 0_R$
Right Zero Divisor
A right zero divisor (in $R$) is an element $x \in R$ such that:
- $\exists y \in R^*: y \circ x = 0_R$
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. 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 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
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): divisor of zero
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): divisor of zero
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): zero divisor