Powers of the Phantom Ideal

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.