A Sheaf Representation for Modules with Applications to Gelfand Rings

Topologies for Primal Ideals

“…As in the case of prime ideals (see, for example, [19, §2.3]) an application of Zorn's lemma shows that minimal primal ideals exist and can be found within any given primal ideal. This was pointed out by H. Simmons during early discussions on the connection between primality and [5] (see [4,Proposition 5.3] for a dual result).…”

Rings of continuous functions with values in a topological division ring

“…Theorem 0.2 is based on Theorem 0.1. Strictly harmonic and Gelfand rings have been studied in [26,51,80,90,97,99]. Now we consider a semitopological division ring, i.e., a topological ring F is a division ring from the algebraic viewpoint.…”

Sheaf representations of strongly harmonic rings

“…For quite different reasons, Borceux and Van den Bossche [5], [6] introduced two sheaf representations for arbitrary rings, and then in [4] it was shown that for Gelfand rings the more interesting of these two representations agrees with the Mulvey-Koh representation. At the time it was not clear just how the more general Koh representation (for strongly harmonic rings) related to either of the Borceux-Van den Bossche representations.…”

“…Sections 1-4 are concerned with various aspects of these general constructions. Section 5 contains a characterization of strongly harmonic rings Harold Simmons analogous to that of Gelfand rings given in [4], and finally in Section 6 we retrieve the Koh representation from our general constructions. §1.…”

“…The second example is due to Borceux and Van den Bossche, and is described in detail in [4]. Following the notation and terminology of [ To construct a sheaf representation of R on the base space S, we must obtain (either directly or indirectly) an indexed family of ring morphism R ->R(p) (for p&S).…”