\begin{definition}[Definition:Proper Zero Divisor]
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.
\end{definition}