2017/12/22 by Sagarmoy Bag, Bag, Sagarmoy, Sudip Kumar Acharyya +3
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1712.08312
openalex publication_date 2017/12/22 · openalex created_date 2022/09/04 · openalex updated_date 2026/07/28
A proper ideal I in a commutative ring with unity is called a\nz^\∘-ideal if for each a in I, the intersection of all minimal prime\nideals in R which contain a is contained in I. For any totally ordered\nfield F and a completely F-regular topological space X, let C(X,F) be\nthe ring of all F-valued continuous functions on X and B(X,F) the\naggregate of all those functions which are bounded over X. An explicit\nformula for all the z^\∘-ideals in A(X,F) in terms of ideals of closed\nsets in X is given. It turns out that an intermediate ring A(X,F)\≠\nC(X,F) is never regular in the sense of Von-Neumann. This property further\ncharacterizes C(X,F) amongst the intermediate rings within the class of\nPF-spaces X. It is also realized that X is an almost PF-space if and\nonly if each maximal ideal in C(X,F) is z^\∘-ideal. Incidentally this\nproperty also characterizes C(X,F) amongst the intermediate rings within the\nfamily of almost PF-spaces.\n