  This paper focuses mainly on the ring of all bounded Baire one functions on a
topological space. The uniform norm topology arises from the $\sup$-norm
defined on the collection $B_1^*(X)$ of all bounded Baire one functions. With
respect to this topology, $B_1^*(X)$ is a topological ring. It is proved that
under uniform norm topology, the set of all units forms an open set and as a
consequence of it, every maximal ideal of $B_1^*(X)$ is closed in $B_1^*(X)$
with uniform norm topology. Since the natural extension of uniform norm
topology on $B_1(X)$, when $B_1^*(X) \neq B_1(X)$, does not show up these
features, a topology called $m_B$-topology is defined on $B_1(X)$ suitably to
achieve these results on $B_1(X)$. It is proved that the relative $m_B$
topology coincides with the uniform norm topology on $B_1^*(X)$ if and only if
$B_1(X) = B_1^*(X)$. Moreover, $B_1(X)$ with $m_B$-topology is 1st countable if
and only if $B_1(X) = B_1^*(X)$. \\ The last part of the paper establishes a
correspondence between the ideals of $B_1^*(X)$ and a special class of
$Z_B$-filters, called $e_B$-filters on a normal topological space $X$. It is
also observed that for normal spaces, the cardinality of the collection of all
maximal ideals of $B_1(X)$ and those of $B_1^*(X)$ are the same.
