  In this paper, we prove that: (1) Let $f:G\rightarrow H$ be a continuous
$d$-open surjective homomorphism; if $G$ is an $\mathbb{R}$-factorizabile
paratopological group, then so is $H$. Peng and Zhang's result \cite[Theorem
1.7]{PZ} is improved. (2) Let $G$ be a regular $\mathbb{R}$-factorizable
paratopological group; then every subgroup $H$ of $G$ is
$\mathbb{R}$-factorizable if and only if $H$ is $z$-embedded in $G$. This
result gives out a positive answer to an question of M.~Sanchis and
M.~Tkachenko \cite[Problem 5.3]{ST}.
