  We begin by investigating the class of commutative unital rings in which no
two distinct elements divide the same elements. We prove that this class forms
a finitely axiomatizable, relatively ideal distributive quasivariety, and it
equals the quasivariety generated by the class of integral domains with trivial
unit group. We end the paper by proving a representation theorem that provides
more evidence to the conjecture that B\'ezout monoids describe exactly the
monoids of finitely generated ideals of commutative unital rings with
distributive ideal lattice.
