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Rough X-Sub-Exact Sequence of Rough Modules

Last modified: 2021-11-19

#### Abstract

Discussions related to exact sequences have an important tool in module theory. In its development, the concept of exact

sequences was generalized into U-exact sequences, V-coexact sequences, and X-sub-exact sequences. The concept of X-sub-exact

sequences has been implemented in generalizations of basis and independent modules. On the other hand, Rough Set Theory is a

mathematical tool used to deal with vagueness and uncertainty problems. The development of the concept of algebraic structure

into the rough set theory is very rapid. Several structures have been developed, such as semigroups, groups, rings, modules, exact

sequences in rough set theory. In this research, we construct a rough X-sub-exact sequence of a rough module over a rough ring.

Furthermore, we give some properties related to intersection and union a finite number of rough modules over the rough ring.

sequences was generalized into U-exact sequences, V-coexact sequences, and X-sub-exact sequences. The concept of X-sub-exact

sequences has been implemented in generalizations of basis and independent modules. On the other hand, Rough Set Theory is a

mathematical tool used to deal with vagueness and uncertainty problems. The development of the concept of algebraic structure

into the rough set theory is very rapid. Several structures have been developed, such as semigroups, groups, rings, modules, exact

sequences in rough set theory. In this research, we construct a rough X-sub-exact sequence of a rough module over a rough ring.

Furthermore, we give some properties related to intersection and union a finite number of rough modules over the rough ring.