Construction of the Rough Quotient Modules over the Rough Ring by Using Coset Concepts - Rahma, Aira; Fitriani, Fitriani; Faisol, Ahmad (Journal of Fundamental Mathematics and Applications (JFMA), 2025)
Given an ordered pair $(U, θ)$ where $U$ is the set universe and $θ$ is an equivalence relation on the set $U$ is called an approximation space. The equivalence relation $θ$ is a relation that is reflexive, symmetric, and transitive. If the set $X \subseteq U$, then we can determine the upper approximation of the set $X$, denoted by $\overline{Apr}(X)$, and the lower approximation of the set $X$, denoted by $\underline{Apr}(X)$. The set $X$ is said to be a rough set on $(U, θ)$ if and only if $\overline{Apr}(X)-\underline{Apr}(X) ≠ \emptyset$. A rough set $X$ is a rough module if it satisfies certain axioms. This paper discusses the construction of a rough quotient module over a rough ring using the coset concept to determine its equivalence classes and discusses the properties of a rough quotient module over a rough ring related to a rough torsion module.