Definition:Proper Zero Divisor

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Definition

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


A proper zero divisor of $R$ is an element $x \in R^*$ such that:

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

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


That is, it is a zero divisor of $R$ which is specifically not $0_R$.


The presence of a proper zero divisor in a ring means that the product of two elements of the ring may be zero even if neither factor is zero.

That is, if $R$ has proper zero divisor, then $\struct {R^*, \circ}$ is not closed.


Examples

Proper Zero Divisors of Integer Multiplication Modulo $6$

Consider the multiplicative monoid of integers modulo $6$, defined by its Cayley table:

$\quad \begin{array} {r|rrrrrr} \struct {\Z_6, \times_6} & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \hline \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 \\ \eqclass 1 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \eqclass 2 6 & \eqclass 0 6 & \eqclass 2 6 & \eqclass 4 6 & \eqclass 0 6 & \eqclass 2 6 & \eqclass 4 6 \\ \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 \\ \eqclass 4 6 & \eqclass 0 6 & \eqclass 4 6 & \eqclass 2 6 & \eqclass 0 6 & \eqclass 4 6 & \eqclass 2 6 \\ \eqclass 5 6 & \eqclass 0 6 & \eqclass 5 6 & \eqclass 4 6 & \eqclass 3 6 & \eqclass 2 6 & \eqclass 1 6 \end{array}$


Thus we have:

$\eqclass 2 6 \times \eqclass 3 6 = \eqclass 0 6$

and:

$\eqclass 4 6 \times \eqclass 3 6 = \eqclass 0 6$


Hence in the ring of integers modulo $6$, there are seen to be $3$ proper zero divisors: $\eqclass 2 6$, $\eqclass 3 6$ and $\eqclass 4 6$.


Proper Zero Divisors of Ring of Order 2 Complex Matrices

Consider the ring of square matrices $\struct {\map {\MM_\C} 2, +, \times}$ of order $2$ over the complex numbers $\C$.

We have:

$\begin {pmatrix} 1 & i \\ i & -1 \end {pmatrix} \begin {pmatrix} 1 & i \\ i & -1 \end {pmatrix} = \begin {pmatrix} 0 & 0 \\ 0 & 0 \end {pmatrix}$

demonstrating that $\begin {pmatrix} 1 & i \\ i & -1 \end {pmatrix}$ is a proper zero divisor of $\struct {\map {\MM_\C} 2, +, \times}$.


Also known as

Some authors exclude $0_R$ as a zero divisor and thus refer to a proper zero divisor simply as zero divisor.

Some sources hyphenate: proper zero-divisor.

Some sources use the more precise term proper divisor of zero.


Also see

  • Results about Proper zero divisors can be found here.


Sources

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