A ring R is strongly
harmonic provided that if M_{1},M_{2} are a pair of distinct maximaI modular
ideals of R, then there exist ideals 𝒜 and ℬ such that 𝒜≦̸M_{1},ℬ≦̸M_{2} 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) Rmodule A, let A_{M} = {a ∈ AaRx = 0 for some αj∉M}) and let
Ã = ∪{A∕A_{M}M ∈ℳ(R)}. Define â(M) = a + A_{M} and r(M) = r + O(M) for
every a ∈ A,r ∈ R and m ∈ℳ(R). Then the mapping ξ_{A} : a↦â is a semilinear
isomorphism of A onto Γ(ℳ(R),ℛ(R))—module Γ(ℳ(R),Ã) in the sense that
ξ_{A} is a group isomorphism satisfying ξ_{A}(ar) = âr for every a ∈ A and
r ∈ R.
