  It is proved that if G is a finite group, then the order of G is a proper
upper bound for the phantom number of G. More specifically, if k is a field
whose characteristic divides the order of G, and $\Phi$ is the ideal of phantom
morphisms in the stable category k[G]-$\underline{\rm Mod}$ of modules over the
group algebra k[G], then $\Phi^{n-1} = 0,$ where n is the nilpotency index of
the Jacobson radical J of k[G]. If $R$ is a semiprimary ring, with $J^n =0,$
and $\Phi$ denotes the phantom ideal in the module category R-Mod, then
$\Phi^n$ is the ideal of morphisms that factor through a projective module. If
R is a right coherent ring and every cotorsion left R-module has a coresolution
of length $n$ by pure injective modules, then $\Phi^{n+1}$ is the ideal of
morphisms that factor through a flat module.
  These results are obtained by introducing the mono-epi (ME) exact structure
on the morphisms of an exact category (${\mathcal A}; {\mathcal E}$), used to
prove new versions of Salce's Lemma, the Ghost Lemma of Christensen, and
Wakamatsu's Lemma. Salce's Lemma gives a bijective correspondence between
special precovering ideals and special preenveloping ideals. The exact category
(Arr (${\mathcal A}$); ME) of morphisms allows us to introduce the notion of an
extension $i \star j$ of morphisms and the notion of an extension of ideals in
(${\mathcal A}; {\mathcal E}$). The Ghost Lemma asserts that the class of
special precovering (resp., special preenveloping) ideals is closed under
products and extensions and that the bijective correspondence of Salce's Lemma
replaces multiplication with extension. Wakamatsu's Lemma is the statement that
if a covering ideal is closed under extensions of morphisms, then it is a
special precovering ideal with a syzygy ideal generated by objects.
