Definition:Proper Zero Divisor

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

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


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

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

Some sources do not insist on $x$ itself being non-zero, that is, zero itself is included in the set of zero divisors.

In this case, the term proper zero divisor is used to define what we call a zero divisor.