There is a binary relation $\mid^*$ defined on any commutative ring as follows: $a \mid^* b$ iff $ak=b$ for some $k \in R$ that is not a zero divisor. This is always transitive, and it is reflexive except on the trivial ring. We also have: $$(*) \qquad x \mid^* 0 \rightarrow x=0.$$
Now in an integral domain, the relation $\mid^*$ almost completely agrees with the more usual $\mid$, with the exception that whereas $x \mid 0$ is always true, nonetheless by $(*)$ the statement $x \mid^* 0$ is never true for $x \neq 0$.
However in other commutative rings it can behave differently. For a simple example, observe that whereas $1 \mid x$ is always true, nonetheless $1 \mid^* x$ is equivalent to $x$ not being a zero-divisor. For another example of how they can diverge, observe that in $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}$, we have $(1,1,0) \mid (1,0,0)$. But it can't be the case that $(1,1,0) \mid^* (1,0,0)$. In general, I think that $\mid^*$ behaves much more like an equivalence relation than $\mid$ does. Its almost as if $a \mid^* b$ is 25% of the way to saying that there exists a unit $u$ such that $au=b$.
Soft Question. Does the relation $\mid^*$ have any interesting applications for understanding the structure of commutative rings that aren't integral domains? If so, what are some examples of not-entirely trivial applications of $\mid^*$?