Direct-sum representations of injective modules

Decompositions of injective modules

“…In the case where M is a direct sum of countably generated indecomposable injective modules, this is contained in results of Faith and Walker [2].…”

“…Let Ml = S Π M t . Then we claim M = Let φ be the natural map from M to M/S, and let ψ be the restriction of φ to D. Condition (2) above implies that ψ is a monomorphism, so im (ψ) is an injective subobject of M/S. We need only show that im(τ/r) is essential, which is an easy consequence of Lemma 1.…”

“…For injective modules over Noetherian rings, ordinary direct sums yield excellent results, due primarily to Matlis [7]. In contrast, Faith and Walker [2] have shown that if R is a non-Noetherian ring, there does not exist any set of injective modules such that any injective module can be imbedded in a direct sum of modules isomorphic to members of this set. In the spectral category, however, reasonable decompositions always exist (Theorem 2 below).…”

“…To show such an S exists note that (i) enables us to apply Zorn's lemma and (iii) guarantees that an ascending union of subobjects satisfying (2) still satisfies (2). Let Ml = S Π M t .…”

Admissible modules and a characterization of reduced left artinian rings

“…In Theorem 3.6 the conditions (1) and (4) can be replaced by conditions (1*) and (4*) which originate from (1) and (4) if we replace JV* by T. Because of 3.6(5) and ΛΓ* g T it is clear that (1*) follows from (1). Likewise (4*b) and (4*c) follow immediately from (1*) by means of [5,Th. 3.3].…”