In the present paper we prove that, for a BCI-algebra
$X$ with condition $(S)$, if the $p$-semisimple part of $X$
is a subalgebra of $X$ then $X$ is isomorphic to direct
product of a BCK-algebra with condition $(S)$ and a
$p$-semisimple BCI-algebra, and obtain other results
on such algebras. Moreover we show that the concepts
of regular ideals, closed $p$-ideals and stong ideals coincide.