The author discusses five different sheaf representations of a ring R with identity. Their relationships are then examined in the context of strongly harmonic rings. If S is a subspace of Spec(R), \({\mathcal O}(S)\) denotes the frame of open subsets of S and if Idl(R) is the lattice of 2- sided ideals of R, then the adjunction Idl(R)\(\rightleftarrows^{d}_{h}{\mathcal O}(S)\) (d\(\dashv h)\) induces two representations of R. On the other hand, given a subframe \(\Omega\) (necessarily compact and spatial) of Idl(R), the adjunction \(\Omega\) \(\rightleftarrows^{i}_{w}Idl(R)\) gives rise to two further representations. A fifth representation is obtained by considering the sheaf on \(\Omega\) which assigns to \(L\in \Omega\) the endomorphism ring \(Hom_ R(L,L).\)
This last representation was considered by F. Borceux and G. van den Bossche [Algebra in a localic topos with applications to ring theory (Lect. Notes Math. 1038) (1983; Zbl 0522.18001)] in the case where \(\Omega\) is the frame of virginal ideals of R. (In the above reference, these ideals are referred to as ”pure” rather than virginal.) An ideal I of R is virginal if and only if \(I+Ann(a)=R\) for all \(a\in R\). This definition is strengthened by the author to \(I+Ann(aR)=R\) for all \(a\in R\), and such an ideal I is called uniformly virginal. These ideals form a subframe \(\Psi\) of Idl(R).
A ring R is strongly harmonic if and only if for distinct maximal ideals M,N of R, there exist a,b\(\in R\) with \(a\not\in N\), \(b\not\in M\) and \(aRb=0\). The author characterizes these rings by the property that the map Idl(R)\(\to^{w}\Psi\) of taking the uniformly virginal part of an ideal has a right adjoint. The above discussion of representations is applied to the case \(\Omega =\Psi\) and \(S=\max R\), the maximal spectrum of R. If R is strongly harmonic, it is shown that the maximal spectrum and the uniformly virginal spectrum are homemorphic and that four of the five sheaf representations coincide. Many of these ideas were investigated for Gelfand rings in the above mentioned work of Borceux and van den Bossche as well as in their collaboration with the author [Proc. Lond. Math. Soc., III. Ser. 48, 230-246 (1984; Zbl 0536.18003)].
The paper under review is clearly written and essentially self contained modulo some acquaintance with sheaf theory. It includes a section on subframes of Idl(R) as well as a section on sheaf representations where the stalks are quotient rings. The generality of the presentation allows one to consider many well-known representations as special cases of these constructions.