Definition:Zero Divisor of Ring

Definition

Let $\left({R, +, \circ}\right)$ be a ring.

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

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

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

The expression:

$x$ is a zero divisor

can be written:

$x \mathop \backslash 0_R$

The conventional notation for this is $x \mid 0_R$, but there is a growing trend to follow the notation above, as espoused by Knuth et al.

Also known as

Some sources hyphenate, as: zero-divisor.

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:

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

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

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

• Proper Zero Divisor

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.2$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 21$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.3$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.3$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 55 \ (5)$