On Goldman's primary decomposition

Extension groups between atoms and objects in locally noetherian Grothendieck category

“…As in [Sto72], the corresponding atomic object H α is the residue field k(p) of p. Hence the residue field k(α) ∼ = End R (k(p)) is also k(p). In general, however, for an atom α in A, the atomic object H α corresponding to α and the residue field k(α) of α do not necessarily coincide with each other.…”

“…We recall the definition of atoms, which was introduced by Storrer [Sto72]. The following properties about monoform objects are well known (for example, [Sto72]).…”

“…The following properties about monoform objects are well known (for example, [Sto72]). For complete proofs, we refer to [Kan12].…”

“…In the case where R is a commutative noetherian ring, the map p → R/p is a bijection between the prime spectrum Spec R of R and ASpec(Mod R). For the details of these arguments, see [Sto72] or [Kan12].…”

“…For a ring R, Storrer [Sto72] introduced the notion of atoms in the category Mod R of right R-modules, which are equivalence classes of monoform modules. In the case where R is a right noetherian ring, it is known that the atoms bijectively correspond to the isomorphism classes of indecomposable injective modules.…”