Definition:Zero Divisor/Also defined as

Zero Divisor: Also defined as

Some sources define a zero divisor as an element $x \in R_{\ne 0_R}$ such that:

$\exists y \in R_{\ne 0_R}: x \circ y = 0_R$

where $R_{\ne 0_R}$ 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 a proper zero divisor.

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 55$. Special types of ring and ring elements: $(5)$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): zero divisors
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): divisor of zero
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): zero divisors
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): zero-divisor