  A space $X$ is called a $k_{R}$-space, if $X$ is Tychonoff and the necessary
and sufficient condition for a real-valued function $f$ on $X$ to be continuous
is that the restriction of $f$ to each compact subset is continuous. In this
paper, we discuss the $k_{R}$-property of products of sequential fans and free
Abelian topological groups by applying the $\kappa$-fan introduced by Banakh.
In particular, we prove the following two results:
  (1) The space $S_{\omega_{1}}\times S_{\omega_{1}}$ is not a $k_{R}$-space.
  (2) The space $S_{\omega}\times S_{\omega_{1}}$ is a $k_{R}$-space if and
only if $S_{\omega}\times S_{\omega_{1}}$ is a $k$-space if and only if
$\mathfrak b>\omega_1$.
  These results generalize some well-known results on sequential fans.
Furthermore, we generalize some results of Yamada on the free Abelian
topological groups by applying the above results. Finally, we pose some open
questions about the $k_{R}$-spaces.
