Topological Rings and Modules Via Operations

The structure of an α(β,β) -topological ring is richer in comparison with the structure of an α(β,β) -topological group. The theory of α(β,β) -topological rings has many common features with the theory of α(β,β) -topological groups. Formally, the theory of α(β,β) -topological abelian groups is included in the theory of α(β,β) -topological rings. The purpose of this paper is to introduce and study the concepts of α(β,β) -topological rings and α(β,γ) -topological R-modules. we show how they may be introduced by specifying the neighborhoods of zero, and present some basic constructions. We provide fundamental concepts and basic results on α(β,β) -topological rings and α(β,γ) -topological Rmodules.