The Properties of Rough Submodule over Rough Ring

A pair of non-empty set $U$ and equivalence relation $R$ on $U$, denoted as $(U,R)$, is called approximation space. Furthermore, equivalence classes form the construction of lower approximation and upper approximation. Let $X\subseteq U$, the lower approximation of $X$ denoted by $\underline{X}$ and the upper approximation of $X$ denoted by $\overline{X}$. A pair $Apr(X)=(\underline{X},\overline{X})$ is a rough set if $\underline{X}\neq \overline{X}$. $Apr(X)$ is rough module if $Apr(X)$ satisfies some conditions. In this research, we investigate some characteristics of the rough module and rough submodule over rough ring. Furthermore, we construct examples of the rough module and the rough submodule on approximation space $(U,R)$.