On α-regular archimedean f-rings

Abstract

Abstract For a commutative ring A with identity, and for infinite cardinals α as well as the symbol ∞, which indicates the situation in which there are no cardinal restrictions, one defines A to be α-regular if for each subset D of A, with |D| < α and de = 0, for any two distinct d, e ∈ D, there is an s ∈ A such that d 2 s = d, for each d ∈ D, and if xd = 0, for each d ∈ D, then xs = 0 This paper studies α-regular archimedean f-rings, relative to lateral α-completeness. The main result is that the operator l(α) that gives the lateral α-completion commutes with b, the reflection that closes an f-ring with respect to bounded inversion. An f-ring is α-regular if and only if it has bounded inversion and is laterally α-complete, and the operator that creates the α-regular hull is r(α) = b · l(α). It is shown that the space mr(α)A of all maximal ℓ-ideals of r(α)A is the same as that of the α-projectable hull. Finally, r(α)A contains the ring of α-quotients, and necessary conditions are given for them to coincide.