ON GENERALIZED COMMUTATIVE RINGS AND RELATED STRUCTURES

Abstract

The study of commutative rings originated from the theory of algebraic number and algebraic geometry [3]. In 1921, Emmy Noether gave the first axiomatic foundation of the theory of commutative rings in her monumental paper “Ideal theory in rings”. The genesis of the theory of commutative rings dates back to the early 19th century while its maturity was achieved in third decade of 20th For any type of abstract algebra, a generalization is a defined class of such algebra. Of course, a generalization of a concept is an extension of the concept to less specific criteria. Generalization plays a vital role to enhance a mathematical concept and to walk around the tracks which leads to achieve new goals. century. Since the introduction of the concept of commutative rings, much progress has been made by many researchers in the development of this concept through generalization [28, 29, 39]. In 1972, a generalization of a commutative semigroup has been introduced by Kazim and Naseeruddin [26]. In ternary commutative law, , they introduced braces on the left of this law and explored a new pseudo associative law, that is . This they called the left invertive law. A groupoid satisfying the left invertive law is called a left almost semigroup and is abbreviated as LA-semigroup. Despite the fact that the structure is nonassociative and non-commutative, however it possesses properties which usually come across in associative and commutative algebraic structures. In 1978, Mushtaq and Yusuf [50] established some interesting and useful results in the theory of LA-semigroups. In [51], they introduced locally associative LAsemigroups. Later in 1991, Mushtaq and Iqbal [43] did an extensive study in decomposition of locally associative LA-semigroup. One can see that Mushtaq and his associates have a remarkable contribution to strengthen and explore new tracks in the theory of LA-semigroups. In 1994, Protic and Stevanovic [56] have used the name Abel-Grassmann’s groupoids(abbreviated as AG-groupoids ), instead of LA-semigroups. For more ii

details of their spad work regarding AG*-groupoids and AG**-groupoids, we refer [57, 58, 59, 69]. Mushtaq and Khan [46, 47, 48, 49] also focused on AG*groupoids and AG**-groupoids and have done extensive work. In 1996, Mushtaq and Kamran [45] extended this concept to left almost group (abbreviated as LA-group). An LA-semigroup is called an LA-group if (i) there exist a left identity such that for all , and (ii) for every there exists a left inverse ∈ G such that For more details see [25]. The fundamental questionnaire like, Why’s, What’s, If’s, always insist the human mind to explore an unending task, to move from known to unknown and from visible to invisible. The passionate purposefulness has always led to the opening of new outlook of expanse. Keeping in view the fundamental questionnaire, we thought of a purpose to generalize commutative rings by an outcome of LA-semigroups and LA-groups. Actually, this offshoot is a non-associative and a non-commutative structure, known as left almost rings (abbreviated as LA-rings), introduced by Yusuf [70]. Left almost rings are in fact a generalization of commutative rings and carries attraction due to its peculiar characteristics and structural formation. This thesis comprises 8 chapters and on the whole it is a threefold study. In first phase we discuss AG-groupoids and Γ-AG-groupoids which are in fact single (binary) operational structures. These structures have been investigated in chapters 2 and 5. In second phase, we deal with two (binary) operational structures such as LA-rings and Γ-LA-rings. These concepts have been studied in chapters 3, 4 and 6. While in the third phase, we look into the application side and investigated the fuzzy concepts of these algebraic structures.