H. Ibrahim, A. Khalaf,
Volume 16, Issue 2 (10-2021)
Abstract
The structure of an $alpha_{(beta, beta)}$-topological ring is richer in comparison with the structure of an $alpha_{(beta, beta)}$-topological group. The theory of $alpha_{(beta, beta)}$-topological rings has many common features with the theory of $alpha_{(beta, beta)}$-topological groups. Formally, the theory of $alpha_{(beta, beta)}$-topological abelian groups is included in the theory of $alpha_{(beta, beta)}$-topological rings.
The purpose of this paper is to introduce and study the concepts of $alpha_{(beta, beta)}$-topological rings and $alpha_{(beta, gamma)}$-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 $alpha_{(beta, beta)}$-topological rings and $alpha_{(beta, gamma)}$-topological $R$-modules.
