Rings and sheaves
This paper is a sequel to the author's earlier survey [Itogi Nauki Tekh., Ser. Algebra, Topologiya, Geom. 29, 119-191 (1991; Zbl 0835.54019)] whose goal is the representation of algebraic objects as sets of functions. Chapter 1 is a historical essay on some of the main theorems in rings of continuous functions, including those of Gelfand-Kolmogorov, Gelfand-Shilov, and Stone-Weierstrass. He also discusses ideals and subrings, including those of Gelfand and Wallman, as well as duality theorems and the Stone property. Chapter 2 is a historical essay and survey of sheaves of rings, leading to sheaves of rings of germs of continuous functions, as well as the spectra of rings. Chapter 3 treats the general theory of sheaf representations of rings, including those of Pierce and Lambek, among others. The survey ends with a bibliography of 550 items.
representation of algebraic objects as sets of functions
scientific article; zbMATH DE number 877634
scientific article