Iranian Journal of

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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.

Keywords: Operations, $alpha_{beta}$-Open set, Rins, $alpha_{(beta, beta)}$-Topological rings, $alpha_{(beta, gamma)}$-Topological $R$-modules

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