Definition:Proper Zero Divisor
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 divisors, then $\struct {R^*, \circ}$ is not closed.
Also known as
Some authors exclude $0_R$ as a zero divisor and thus refer to this concept simply as zero divisor.
Some sources use the more precise term proper divisor of zero.
Examples
Proper Zero Divisors of Integer Multiplication Modulo $6$
Consider the multiplicative monoid of integers modulo $6$, defined by its Cayley table:
- $\begin{array} {r|rrrrrr} \struct {\Z_6, \times_6} & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \hline \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 \\ \eqclass 1 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \eqclass 2 6 & \eqclass 0 6 & \eqclass 2 6 & \eqclass 4 6 & \eqclass 0 6 & \eqclass 2 6 & \eqclass 4 6 \\ \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 \\ \eqclass 4 6 & \eqclass 0 6 & \eqclass 4 6 & \eqclass 2 6 & \eqclass 0 6 & \eqclass 4 6 & \eqclass 2 6 \\ \eqclass 5 6 & \eqclass 0 6 & \eqclass 5 6 & \eqclass 4 6 & \eqclass 3 6 & \eqclass 2 6 & \eqclass 1 6 \\ \end{array}$
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 see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 3$. Definition of an Integral Domain
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $3$: Some special classes of rings
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): divisor of zero
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): proper divisors of zero
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 1$: Exercise $9$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): divisor of zero
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): proper divisors of zero