TY - JOUR JF - IJMSI JO - IJMSI VL - 16 IS - 2 PY - 2021 Y1 - 2021/10/01 TI - Topological Rings and Modules Via Operations TT - N2 - 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. SP - 31 EP - 48 AU - Ibrahim, H. AU - Khalaf, A. AD - Department of Mathematics, Faculty of Education, University of Zakho KW - Operations KW - $alpha_{beta}$-Open set KW - Rins KW - $alpha_{(beta KW - beta)}$-Topological rings KW - $alpha_{(beta KW - gamma)}$-Topological $R$-modules UR - http://ijmsi.ir/article-1-1208-en.html DO - 10.52547/ijmsi.16.2.31 ER -