On a representation of a strongly harmonic ring by sheaves.

A ring R is strongly harmonic provided that if Mu M2 are a pair of distinct maximal modular ideals of R, then there exist ideals Stf and & such that Szf % Mu & % M2 and S/έ% — 0. Let ^^(JR) be the maximal modular ideal space of R. If Me ΛT{R\ let 0{M) = {reR\for some y&M, rxy = 0 for every xeR}. Define &(JR) = U {RI0(M) \ Me^f(R)}. If i£ is a strongly harmonic ring with 1, then R is isomorphic to the ring of global sections of the sheaf of local rings &(R) over ^(R). Let Γ^{R),&{R)) be the ring of global sections of &(JR) over ^#(R). For every unitary (right) iίί-module A, let AM = {a e A | aRx = 0 for some x£M) and let A \J{AIAM I Me ̂ T(S)}. Define ά{M) = a + AM and r(ikΓ) = r -f O(Jlf) for every aeA,reR and m e ^(R). Then the mapping f A: α i-̂ a is a semi-linear isomorphism of A onto ΓθT(i?)), ^CR))—module Γ(^f(R), A) in the sense that fA is a group isomorphism satisfying ?A(α?) = ar for every α e A and reR.