Definition:Proper Zero Divisor
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Definition
Let $\left({R, +, \circ}\right)$ 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 $R \setminus \left\{{0_R}\right\}$.
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 $\left({R^*, \circ}\right)$ is not closed.
Also see
Sources
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 21$
