A study of roughness in different algebraic structures

Abstract

Groups, rings and modules are the fundamental concepts of abstract algebra. The study of groups started in 18th century and gradually developed in various decades, initiated by very famous mathematicians Cauchy, Abel, Galois and Lagrange. The other fundamental concepts of abstract algebra (e.g. rings and modules) are the generalizations of group. Rings and modules are the central notions of Commutative algebra and Homological algebra. The development of these mathematical theories has been greatly influenced by numerous problems and ideas occurring naturally in algebraic number theory and algebraic geometry, algebraic topology. Many books have been written on these fundamental concepts. For systematic evaluation of groups, rings and modules, we refer the reader to [14, 5]. However, most of the classical mathematical methods and solutions do not address the imprecision and uncertainties in given information. A genius mind L. A. Zadeh said: ”The closer one looks at a real world problem, the fuzzier becomes its solution.” In 1965, Zadeh [46] formulated the concept of fuzzy set theory to describe the vagueness and imprecision containing most of the data. Fuzzy sets handel such situations by attributing a degree to which a certain object belong to a set. Fuzzy sets has a wide range of applications in applied sciences such as computer sciences, management sciences, control sciences, robotics, artificial intelligence, pattern recognition and operation research etc. In its trajectory of stupendous growth, it has also come to include the theory of fuzzy algebra and for the past several decades, several researchers have been working on the fuzzy commutative algebra. Rosenfeld [36] inspired the fuzzification of algebraic structures and introduced the notion of fuzzy subgroups in 1971. Pan defined fuzzy quotient modules and fuzzy exact sequences (see [33]). A systematic description of fuzzy commutative algebra by Mordeson and Malik appeared in [31], and Wang et al. [44] where one can find detailed theoretical study on various algebraic structures.