Any deeper "duality" between non-zero-divisors and units of a ring? I'm reading Aluffi's algebra book at the moment -- specifically, I'm on the introductory rings/modules chapter. 
I noticed two interesting pieces of information: in a (not necessarily commutative) ring $R$,


*

*$x\mapsto ux$ is injective for $u$ a left non-zero-divisor

*$x\mapsto ux$ is surjective for $u$ a left unit
Is there any deeper (read: probably categorical) reason for this apparent "duality"?
Edit: Specifically, since we know that injectivity and surjectivity (well, mono- and epi-ness, but we're in $\sf Set$) are "dual", is there also some duality between the properties of being a left (resp. right) non-zero-divisor and being a left (resp. right) unit?
 A: Here is something which comes to my mind. I don't know if it answers the question properly.
Recall that a monoid (resp. a ring) is the "same" as a category (resp. linear category) with one object. The composition of morphisms is just the multiplication of monoid (resp. ring) elements. The notions of left units ( := right invertible element, what I suspect from the question) and left regular elements ( := left cancellable elements) generalize to arbitrary categories:
A morphism $f : X \to Y$ is left invertible, or more commonly, a split monomorphism, if there is a morphism $g : Y \to X$ such that $g \circ f = \mathrm{id}_X$. A morphism is called left cancellable, or more commonly, an epimorphism, if for all morphisms $g,h : Z \to X$ with $f \circ g = f \circ h$ we have $g=h$. The dual notions are split epimorphism and monomorphism. Then, we have the following:


*

*$f$ is a split monomorphism if and only if $\hom(Y,Z) \to \hom(X,Z),~ g \mapsto g \circ f$ is surjective for all objects $Z$.

*$f$ is a split epimorphism if and only if $\hom(Z,X) \to \hom(Z,Y),~ g \mapsto f \circ g$ is surjective for all objects $Z$.

*$f$ is a monomorphism if and only if $\hom(Z,X) \to \hom(Z,Y),~ g \mapsto f \circ g$ is injective for all objects $Z$.

*$f$ is an epimorphism if and only if $\hom(Y,Z) \to \hom(X,Z),~ g \mapsto g \circ f$ is injective for all objects $Z$.

*Every split monomorphism is a monomorphism.

*Every split epimorphism is an epimorphism.


From here it becomes clear that monomorphisms  and split epimorphisms (similarly, epimorphisms and split monomorphisms) have dual characterizations in terms of $\hom(Z,-)$ (resp. $\hom(-,Z)$). But I doubt that this means that the notions are dual to each other.
