Investigation of a Class of Non-Associative Rings and its Study using Soft Sets

Abstract

One of the end lessly a llu ring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories. Pure mathematics is, in its way the poetry of logica l ideas. Today mathematics especia lly pure mathematics is not the same as it was fifty years ago. Many revolutions have occurred and it has taken new shapes with the due course of time. A major change took place in the mid of 19th century when t he concepts of non-associative rings and non-associative a lgebras were introduced. The theory of non-associative rings a nd algebras has evo lved into a n independ ent branc h of algebra, exhibiting many points of contact with oth er fie lds of mathematics and a lso with physics, mechanics, biology, and other sciences. The centra l part of the theory is the t heory of what are known as nearly-associative rings and a lgebras: Lie, a lternative, Jordan , Ma l'tsev rings and a lgebras, and some of their genera lizations (see Lie a lgebra, Alternative rings and a lgebras, Jordan a lgebra, Mal'tsev a lgebra). In 1921, Emmy Noether gave the first axiomatic foundation of the theory of commutative rings In her monumental paper "Ideal theory in rings". The study of commutative rings originated from the theory of algebraic number and a lgebraic geometry, discussed by M. F. Atiyah in her book Introd uctio n to Commutative Algebra in 1969. T he genesis of t he theory of commutative ri ngs dates back to the early 19th century whi le its maturity was achieved in third deca de of 20th century. For any type of abstract algebra, a generalization is a defined class of such algebra. Of course, a genera lization of a concept is an extension of the concept to less specific criteria. Genera lization plays a vital role to enhance a mathematica l concept and to walk around the tracks which leads to achieve new goals. Since th e introduction of the concept of commutative rings, much progress has been made by many researchers in the development of this concept through genera lization. In 1972, a generalization of a commutative semlgroup has been introduced by Kazim and Naseeruddin . In ternary commutative law, abc = cba, they introduced braces on the left of this law and explored a new pseudo associative law, that is (ab)c = (cb)a. 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 non-associative and non-commutative, however it possesses properties wh ich usually come across in associative and commutative a lgebraic structures. A systematic investigation of left almost semigroups had started since 1978. Further, in 1996 Q. Mushtaq a nd S. Kamran extended the concept of left a lmost groups. The left a lmost group which though is a non-associative structure has interesting resemblance with an abelian group. Due to non-smooth structural formation, many authors estab lished several usefu l results for LAsemigroups and LA-groups. With these inspirations in 2006, the concept of Left a lmost ring was initiated. Left almost rings are in fact a genera lization of commutative rings and carries attraction due to its peculiar characteristics and structural formation . Although the concept of Left a lmost rings was introduced in 2006, but the systematic study of LA-rings goes into the credit of T. Shah and I. Rehman . In current decade, T . Shah and his co-researchers have put forward a remarkable contribution to the development of this particular non-associative and non-commutative structure. For instance, T . Shah and I. Rehman in 2010, utilized the LA-semigroup and LA-ring and genera lized the notion of a commutative semigroup ring. Further in 2010, T. Shah and I. Rehman defined the notion of LA-module over an LA-ring, a non-a beli an non-associative structure but closer to abelian group. Later in 2010, T. Shah et aI., introduced the notion of topological LA-groups and topological LA-rings which are some genera lizations of topological groups and topological rings respectively. They extended some characterizations of topological groups and topologica l rings to topological LA-groups and topological LA-rings. In 2011, M. Shah and T. Shah established some basic and structural facts of LA-ring which will be useful for future research on LA-ring. Also in 2011, T . 11 Shah et aI., promoted the notion of LA-module over an LA-ring defined in and further established the substructures, operations on substructures and quotient of an LA-module by its LA-sub module. They also indicated the non similarity of an LA-module to the usual notion of a module over a commutative ring. Moreover, in 2011, T. Shah, F. Rehman and M. Raees have genera lized the concept of LA-ring by introducing the notion of near left almost ring (abbreviated as nLA-ring). Also, T. Shah, G. Ali and F. Rehman in 2011 characterized nLA-ring t hro ugh its id ea ls. They have shown t hat t he sum of ideals is aga in an idea l, and established t he necessary and sufficient cond ition for an nLA-ring to be direct sum of its idea ls. Furthermore they observed t hat the product of idea ls is just a left idea l. In 2012, T. Shah and I. Rehman explored some notations of idea ls and M-systems in LA-ring. They characterized LA-rings through some properti es of their idea ls. Later T. Shah et a I. , in 2012 have applied the concept of intuitionistic fuzzy sets and established some usefu l results. In 2014, A. M. Alghamdi and F. Sahraoui broaden the concept of LA-module by constructing a tensor product of LA-modules. In addition, P. Yiarayong in 2014 studied left ideals, left primary and weakly left primary ideals in LA-rings. Some characterizations of left primary and weakly left primary ideals were obtained . Recently, in 2015, F. Hussain and W. Khan characterized LA-rings by congruence re lations. They had shown that each homomorphism of LA-rings defines a congruence re lation on LA-rings. In real world there are many problems where trad it ional mathematica l tools are not useful due to uncertainty involved t here. In order to deal with uncertainty theories li ke probabi li ty and interval mathematics have been a pplied since long. Although t hese theories have t heir ap plications yet difficulties are associated with these too. Urge to dea l with uncertainty by tools different from that of probability lead the way to many t heories which have been developed to deal with uncerta in t ies, for example, in 1965 theory of fuzzy sets was introduced, theory of intuitionistic fuzzy sets in 1986, III theory of inte rval mathematics in 1987, t heory of vague sets were given in 1994 a nd t heory of rough sets in 2002. Though many techniques have been developed as a resu lt of these theories, yet difficulties are seem to be there. In order to model vagueness and uncertainties, D. Molodtsov introduced the concept of soft set theory and it has received much attention since its inception. In 1999, he presented the fundamental results of new theory and successfully appli ed it into several directions such as smoothness of functions, game theory, operations research, Riemann-integration, Perron integration, theory of probability etc. A soft set is a coll ection of approximate description of an object. He also showed how soft set theory is free from parameterization inadequacy syndrome of fuzzy set theory, rough set theory, probabi lity theory and game theory. Soft systems provide a very general framework with the involvement of parameters. Research works on soft set theory and its applications in various fields are progressing rapidly in these years. Application of soft set theory in algebraic structures was introduced by Aktas and <;:agman in 2007. They discussed the notion of soft groups and derived some basic properties. They also showed that soft groups extend the concept of fu zzy groups. Jun in 2008, investigated soft BCK/ BCIa lgebra and its application in ideal theory. Feng et aI., in 2008 used the soft set theory to initiate the study of soft semi-rings, soft sub semi-rings, soft ideals, idealistic soft semi-rings and soft semiring homomorphism. In 2008 , Sun et aI., presented the definition of soft modules and constructed some basic properties using modules. In 2010, Acar et aI., introduced t he basic notions of soft rings, which are actually a parameterized family of subrings of a ring, over a ring. In 2010, Babitha et aI., introduced the concept of soft set relations as a sub soft set of the cartesian product of the soft sets and many related concepts. Atagun et aI., in 2011 introduced soft subfields of a fie ld and soft submodul e of a left R-module and investigated their related properties about soft substructures of rings, fields and modules. Further iv in 2011, Celik et aI., introduced the notion of soft ring and soft ideal over a ring and provided some related examples. T. Shah et aI., in 2011, investigated the concept of soft ordered Abel-Grassman 's Groupoids. In 2012, A. Sezgin et aI., introduced the union soft subnear-rings and union soft ideals of a near-ring. X. Liu et aI., in 2012 established some useful fuzzy isomorphism theorems of soft rings. They also introduced the concept of fuzzy ideals of soft rings. If we review the literature of soft set theory, we can observe that the application of soft sets has been restricted to associative algebraic structures (groups, semigroups, associative rings, semi-rings etc). Acceptably, though the study of soft sets, where the base set of parameters is a commutative structure, has attracted the attention of many researchers for more than one decade. But on the other hand there are many sets which are naturally endowed by two compatible binary operations forming a non-associative ring and we may dig out many examples which investigate a non-associative structure in the context of soft sets. Thus it seems natural to apply the concept of soft sets to non-commutative and non-associative structures (LA-rings). To achieve this purpose our focus is LA-rings as it is a newly established non-associative and non-commutative ring structure. Due to peculiar characteristics of this particular structure, certainly it has much scope to work with in it and to develop the theory of soft sets in this structure and eventually this is the main theme of this thesis. This thesis comprises of 9 chapters and on the whole it is a twofold study. In first phase, our main focus is on the classical/theoretical work on LA-rings and LA-modules. This work has been devoted to chapters 2, 3 and 7. While in second phase, we deal with the application of soft set theory to LA-rings and LA-modules. These concepts have been studied in chapters 4, 5, 6 and 8. In chapter 1, we give a brief history of LA-semigroup, LA-group, LA-ring, LA-module, soft sets, soft rings, soft modules and discuss some of their basic properties which are directly related to our study. v In c hapter 2, we present a comprehensive survey and developments of existin g literature of nonassociative rings and enumerate some of their various applications in different directions to date. We do believe that this survey would be un ique in its own way for the reason that such comprehensive and complete information regarding all types of non-associative rings under one umbrella can hard ly be found . We hope that this work will provide an endless source of inspiration for future research in no n-associative ring t heory. He re we tried t o g ive t he literature survey of all no n-associative rings a nd t heir developments in diffe re nt t ime periods includ ing LA-rings (a class of no n-associative rings), recent ly introduced in 2006. It is worth mentioning that althoug h t he concept of LA-rings has been introduced in 2006 by S. M. Yusuf, but t he exist ence of t he structure of t he no n-associative LA-ring has been rema ined in doubt because a ll t he examples given by diffe rent authors so far have assoc iative mult iplica tion . We t ac kl ed this problem a nd gave non-trivi a l exa mples of LA-ring which ca n be fo und in chapter 3. In chapter 3, we prove the existence of non-assoc iative LA-ring a nd LA-field t hrough different non-trivial examples. We also give some computational information about these concepts by providing the cayley tables that we have found by software Mace4. Actually t he purpose of giving t he non-trivial examples of LA-ring in the form of cayley tab les is based to apply the concept of soft set to a lgebraic structure LA-ring which has been disc ussed in chapter 4 and 5. T he contents of chapter 3 have been pu blished in an 15 1 index journal with impact factor 0.333: Analele Stiintifice ale Universitatii Ovidius, Constanta, 21(3), (2013), 223- 228. In cha pte r 4, o ur study is distributed in t o two sections. In section one; we in troduce t he concept soft LA-ri ngs by a pplying t he Mo lodstov's soft set t heory. We do provide number of examp les to ill ustrate t he concepts of soft idea ls, soft prime idea ls a nd soft semi-prime id ea ls. In section two, we cont inued t he developments made in sectio n o ne, regarding t he concept of idea listic soft LAVI rings and establish various results by corresponding examples. Also we investigate the properties of idealistic soft LA-ring with respect to LA-ring homomorphism. The contents of chapter 4 have been accepted in 151 index journal with impact factor 1.812: Journal of Intelligent and Fuzzy Systems, vol. 30, no. 3, (2016), 1537-1546. In chapter 5, in continuation to chapter 4, we have taken a step forward to study the concepts of soft M-system, soft P-system and esta blish some re lated results in soft LA-ring. We make a new approach to apply the Molodstov 's soft set theory to a class of non-associative rings and its ideals. We do provide number of examples to ill ustrate the concepts of soft M-system, soft P-system and soft I-system in soft non-associative-ring. The contents of chapter 5 have been accepted in 15 1 index journal with impact factor 0.405: V.P.B. Sci. Bull., Series A, Vol. 77, Iss. 3, 2015, 131-142. In chapter 6, we extend the notion of soft LA-rings (discussed in chapter 4 and 5) to soft LA-modules. This chapter conta ins two sections. In section one, we define the not ions of soft LA-module, soft subLA-modu le and maximal and minimal soft subLA-modu les and a lso discussed their related results. In section two, we introduce the concepts of quotient soft LA-module, soft LA-homomorphism and consequently estab lish some properties of soft LA-modules regarding exact sequence. Basically chapter 7 has been included in phase one of t he study, in which we establish some theoretical results regarding projective and injective LA-modules. In fact, projective and injective LA-modules are generalization of projective and injective modules. We are with plan to use the concepts of soft sets to projective and injective LA-modules in the subsequent chapter. The scheme of study of this chapter comprises two sections. Section one is devoted to projective LA-modu les, free LA-modu le, split sequence in LA-module and its related results. To construct t he example of projective LA-module we have define LA-vector space also and which intuitively would be the most Vll useful tool for furth er developments. In section two, we define injective LA-modul e a nd prove some of its related results. In chapter 8, we apply the concepts of soft sets to the notions discussed in chapter 7. Here is the right place to use the concepts of soft LA-homomorphism and exact sequence in soft LA-modules as discussed earlier in chapter 6; to develop the theory of projective and injective soft LA-modules. Lastly, in chapter 9 we give conclusion of our study and a lso give some future prospects of th is study.