Vol. 41, No. 2, 1972

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On a representation of a strongly harmonic ring by sheaves

Kwangil Koh

Vol. 41 (1972), No. 2, 459–468

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 such that 𝒜≦̸M1,≦̸M2 and 𝒜ℬ = 0. Let (R) be the maximal modular ideal space of R. If Me(R), let 0(M) = {r R| for some yM,rxy = 0 for every x R}. Define (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) over (R). Let Γ((R),(R)) be the ring of global sections of (R) over (R). For every unitary (right) R-module A, let AM = {a A|aRx = 0 for some αjM}) and let à = ∪{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 : aâ is a semi-linear isomorphism of A onto Γ((R),(R))—module Γ((R),Ã) in the sense that ξA is a group isomorphism satisfying ξA(ar) = âr for every a A and r R.

Mathematical Subject Classification 2000
Primary: 16A64
Secondary: 46J25
Received: 10 February 1971
Published: 1 May 1972
Kwangil Koh