Over a ring R, E denotes
injective hulls, Z singular submodules. General Theorem. For any ring R and anyinfinite regular cardinal σ, conditions (1), (2), and (3) are all equivalent: (1) Anydirect sum of nonsingular right ideals of R contains fewer than σ nonzerosummands. (2) If {Wγ|γ ∈ Γ} is any indexed set of modules with all ZWγ= 0,then the submodule ΠσE(Wγ) = {x = (xγ) ∈ ΠE(Wγ)||supportx < σ} isinfective. (3) For any family {Wγ|γ ∈ Γ} having all ZWγ= 0, the submoduleΠσWγ≤ ΠWγis a complement. Corollary. Every ring R satisfies (1), (2), and (3)for a unique smallest infinite regular cardinal σ = σ(R). Theorem. For anymodule M with ZM = 0, E(M) = C ⊕ D, where C contains no uniformsubmodules and D = Π{E(Dτ)|τ ∈ Ξ}. The submodules C, D, and the Dτare allunique. Each Dτis a direct sum of isomorphic indecomposable injectives allof the same type τ. The cardinal number of such summands of Dτ is the
τ-dimension of M. More general uniform dimensions are constructed for arbitrary
modules.
