AU - Ibrahim, H. AU - Khalaf, A. TI - Topological Rings and Modules Via Operations PT - JOURNAL ARTICLE TA - IJMSI JN - IJMSI VO - 16 VI - 2 IP - 2 4099 - http://ijmsi.ir/article-1-1208-en.html 4100 - http://ijmsi.ir/article-1-1208-en.pdf SO - IJMSI 2 ABĀ  - 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. CP - IRAN IN - Kurdistan Region, Iraq LG - eng PB - IJMSI PG - 31 PT - Research paper YR - 2021