The Spectral Category and Von-neumann Regular Rings

All rings considered are associative with identity and all occurring modules are unital right modules. We denote by Mod R the category of all R-modules. The spectrum of a ring R is known to be the “set” of isomorphism classes of indecomposable injective right R-modules. When R is right-Noetherian the spectrum describes all injective Rmodules, since each injective module is a direct sum of indecomposable submodules. We want to briefly show that for any R (or even for every Grothendieck category), this spectrum can be replaced by the so-called spectral category; one obtains this spectral category by formally inverting all essential monomorphisms. Approaches to such considerations are provided by the work of JOHNSON[6] and UTUMI[8]. As an application one obtains for each module invariant, that the invariant coincides with that which FUCHS[3] introduced under strong assumptions. 1. The Spectral Category of a Grothendieck Category 1.1. Let A be a Grothendieck category, i.e. an abelian category with exact direct limits and a generator. Exactness of direct limits is equivalent to the following statement: (*) For each family (Aλ)λ∈Λ of objects of A B ⊂ ⊕ λ∈Λ Aλ is B = sup Γ ( B ∩ ⊕ λ∈Γ Aλ ) where all finite subsets of Λ factor through Γ. (Often you see (*) as a special case of AB5)[5]. Suppose conversely that (Aλ)λ∈Λ is an increasing filtered family of subobjects of an object A and A′ is another subobject of A. When