  Considering the deeper reasons of the appearance of a remarkable
counterexample by J.~Kaad and M.~Skeide [17] we consider situations in which
two Hilbert C*-modules $M \subset N$ with $M^\bot = \{ 0 \}$ over a fixed
C*-algebra $A$ of coefficients cannot be separated by a non-trivial bounded
$A$-linear functional $r_0: N \to A$ vanishing on $M$. In other words, the
uniqueness of extensions of the zero functional from $M$ to $N$ is focussed. We
show this uniqueness of extension for any such pairs of Hilbert C*-modules over
W*-algebras, over monotone complete C*-algebras and over compact C*-algebras.
Moreover, uniqueness of extension takes place also for any one-sided maximal
modular ideal of any C*-algebra. Such a non-zero separating bounded $A$-linear
functional $r_0$ exist for a given pair of full Hilbert C*-modules $M \subseteq
N$ over a given C*-algebra $A$ iff there exists a bounded $A$-linear
non-adjointable operator $T_0: N \to N$ such that the kernel of $T_0$ is not
biorthogonally closed w.r.t. $N$ and contains $M$. This is a new perspective on
properties of bounded modular operators that might appear in Hilbert C*-module
theory. By the way, we find a correct proof of [13, Lemma 2.4] in the case of
monotone complete and compact C*-algebras, but not in the general C*-case.
