A ring R is strongly
harmonic provided that if M1,M2 are a pair of distinct maximaI modular
ideals of R, then there exist ideals 𝒜 and ℬ such that 𝒜≦̸M1,ℬ≦̸M2 and
𝒜ℬ = 0. Let ℳ(R) be the maximal modular ideal space of R. If Meℳ(R),
let 0(M) = {r ∈ R| for some y∉M,rxy = 0 for every x ∈ R}. Define
ℛ(R) = ∪{R∕O(M)|M ∈ℳ(R)}. If R is a strongly harmonic ring with 1, then R is
isomorphic to the ring of global sections of the sheaf of local rings ℛ(R) over ℳ(R).
Let Γ(ℳ(R),ℛ(R)) be the ring of global sections of ℛ(R) over ℳ(R). For every
unitary (right) R-module A, let AM= {a ∈ A|aRx = 0 for some αj∉M}) and let
Ã= ∪{A∕AM|M ∈ℳ(R)}. Define â(M) = a + AM and r(M) = r + O(M) for
every a ∈ A,r ∈ R and m ∈ℳ(R). Then the mapping ξA: a↦â is a semi-linear
isomorphism of A onto Γ(ℳ(R),ℛ(R))—module Γ(ℳ(R),Ã) in the sense that
ξA is a group isomorphism satisfying ξA(ar) =âr for every a ∈ A and
r ∈ R.
