  Inspired by the Douglas lemma, we investigate the solvability of the operator
equation $AX=C$ in the framework of Hilbert C*-modules. Utilizing partial
isometries, we present its general solution when $A$ is a semi-regular
operator. For such an operator $A$, we show that the equation $AX=C$ has a
positive solution if and only if the range inclusion ${\mathcal R}(C) \subseteq
{\mathcal R}(A)$ holds and $CC^*\le t\, CA^*$ for some $t>0$. In addition, we
deal with the solvability of the operator equation $(P+Q)^{1/2}X=P$, where $P$
and $Q$ are projections. We provide a counterexample to show that there exists
a $C^*$-algebra $\mathfrak{A}$, a Hilbert $\mathfrak{A}$-module $\mathscr{H}$
and projections $P$ and $Q$ on $\mathscr{H}$ such that the operator equation
$(P+Q)^{1/2}X=P$ has no solution. Moreover, we give a perturbation result
related to the latter equation.
