Nonassociative Rings Research Articles

Alternative Derivations in (1,1) Rings

Objectives: To show the associativity in one of the subclass of non-associative (γ,δ) rings. Method: Derivation alternator rings are limiting case of associative rings. The ring (1,1) is one of the sub class of (γ,δ) rings. Consider a (1,1) derivation alternator ring R, with characteristic ≠ 2, and it is well known that this ring is neither alternative nor flexible. In this paper it will be proved that right alternative property (R,x,x) holds in R and flexibility follows, finally associativity arrives in R. Findings: If this ring R does not contain nilpotent elements even though it will be associative. Novelty: Further investigators may extend the applications of these rings in science and engineering fields. Mathematics Subject Classification: 2010 MSC 17D

Nonassociative Algebras, Rings and Modules over Them

The review is devoted to nonassociative algebras, rings and modules over them. The main actual and recent trends in this area are described. Works are reviewed on radicals in nonassociative rings, nonassociative algebras related with skew polynomials, commutative nonassociative algebras and their modules, nonassociative cyclic algebras, rings obtained as nonassociative cyclic extensions, nonassociative Ore extensions of hom-associative algebras and modules over them, and von Neumann finiteness for nonassociative algebras. Furthermore, there are outlined nonassociative algebras and rings and modules over them related to harmonic analysis on nonlocally compact groups, nonassociative algebras with conjugation, representations and closures of nonassociative algebras, and nonassociative algebras and modules over them with metagroup relations. Moreover, classes of Akivis, Sabinin, Malcev, Bol, generalized Cayley–Dickson, and Zinbiel-type algebras are provided. Sources also are reviewed on near to associative nonassociative algebras and modules over them. Then, there are the considered applications of nonassociative algebras and modules over them in cryptography and coding, and applications of modules over nonassociative algebras in geometry and physics. Their interactions are discussed with more classical nonassociative algebras, such as of the Lie, Jordan, Hurwitz and alternative types.

Splitting theorems for nonassociative rings

Splitting theorems for nonassociative rings

On the commutativity of non-associative rings

On the commutativity of non-associative rings

Satisfiability in MultiValued Circuits

Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras $\mathbf A$ there is a polynomial time algorithm that decides if an equation over $\mathbf A$ has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time.

Some Developments in the Field of Homological Algebra by Defining New Class of Modules over Nonassociative Rings

The LA‐module is a nonassociative structure that extends modules over a nonassociative ring known as left almost rings (LA‐rings). Because of peculiar characteristics of LA‐ring and its inception into noncommutative and nonassociative theory, drew the attention of many researchers over the last decade. In this study, the ideas of projective and injective LA‐modules, LA‐vector space, as well as examples and findings, are discussed. We construct a nontrivial example in which it is proved that if the LA‐module is not free, then it cannot be a projective LA‐module. We also construct free LA‐modules, create a split sequence in LA‐modules, and show several outcomes that are connected to them. We have proved the projective basis theorem for LA‐modules. Also, split sequences in projective and injective LA‐modules are discussed with the help of various propositions and theorems.

From Binary Groups to Terminal Rings

Binary groups are a meaningful step up from non-associative rings and nearrings. It makes sense to study them in terms of their nearrings of zero-fixing polynomial maps. As this involves algebras of a more specialized nature these are looked into in sections three and four. One of the main theorems of this paper occurs in section five where it is shown that a binary group V is a P0(V) ring module if, and only if, it is a rather restricted form of non-associative ring. Properties of these non-associative rings (called terminal rings) are investigated in sections six and seven. The finite case is of special interest since here terminal rings of odd order really are quite restricted. Sections eight to thirteen are taken up with the study of terminal rings of order pn (p an odd prime and n ≥ 1 an integer ≤ 7).

On Non-Associative Rings

Jordan ring is one example of the non-associative rings. We can construct a Jordan ring from an associative ring by defining the Jordan product. In this paper, we discuss the properties of non-associative rings by studying the properties of the Jordan rings. All of the ideals of a non-associative ring R are non-associative, except the ideal generated by the associator in R. Hence, a quotient ring <img src=image/13422591_01.gif> can be constructed, where <img src=image/13422591_02.gif> is the ideal generated by associators in R. The fundamental theorem of the homomorphism ring can be applied to the non-associative rings. By a little modification, we can find that <img src=image/13422591_01.gif> is isomorphic to <img src=image/13422591_03.gif>. Furthermore, we define a module over a non-associative ring and investigate its properties. We also give some examples of such modules. We show if M is a module over a non-associative ring R, then M is also a module over <img src=image/13422591_01.gif> if <img src=image/13422591_02.gif> is contained in the annihilator of R. Moreover, we define the tensor product of modules over a non-associative ring. The tensor product of the modules over a non-associative ring is commutative and associative up to isomorphism but not element by element.

IND-CCA1 Secure FHE on Non-Associative Ring

A fully homomorphic encryption (FHE) would be the important cryptosystem as the basic scheme for the cloud computing. Since Gentry discovered in 2009 the first fully homomorphic encryption scheme, some fully homomorphic encryption schemes were proposed. In the systems proposed until now the bootstrapping process is the main bottleneck and the large complexity for computing the ciphertext is required. In 2011 Zvika Brakerski et al. proposed a leveled FHE without bootstrapping. But circuit of arbitrary level cannot be evaluated in their scheme while in our scheme circuit of any level can be evaluated. The existence of an efficient fully homomorphic cryptosystem would have great practical implications in the outsourcing of private computations, for instance, in the field of the cloud computing. In this paper, IND-CCA1secure FHE based on the difficulty of prime factorization is proposed which does not need the bootstrapping and it is thought that our scheme is more efficient than the previous schemes. In particular the computational overhead for homomorphic evaluation is O(1).

Non-Associative Algebra Redesigning Block Cipher with Color Image Encryption

The substitution box (S-box) is a fundamentally important component of symmetric key cryptosystem. An S-box is a primary source of non-linearity in modern block ciphers, and it resists the linear attack. Various approaches have been adopted to construct S-boxes. S-boxes are commonly constructed over commutative and associative algebraic structures including Galois fields, unitary commutative rings and cyclic and non-cyclic finite groups. In this paper, first a non-associative ring of order 512 is obtained by using computational techniques, and then by this ring a triplet of 8 × 8 S-boxes is designed. The motivation behind the designing of these S-boxes is to upsurge the robustness and broaden the key space due to non-associative and non-commutative behavior of the algebraic structure under consideration. A novel color image encryption application is anticipated in which initially these 3 S-boxes are being used to produce confusion in three layers of a standard RGB image. However, for the sake of diffusion 3D Arnold chaotic map is used in the proposed encryption scheme. A comparison with some of existing chaos and S-box dependent color image encryption schemes specs the performance results of the anticipated RGB image encryption and observed as approaching the standard prime level.

FURTHER REMARKS ON ELEMENTARY RADICALS AND ASSOCIATED FILTERS OF IDEALS

AbstractRyabukhin showed that there is a correspondence between elementary radical classes of rings and certain filters of ideals of the free ring on one generator, analogous to the Gabriel correspondence between torsion classes of left unital modules and certain filters of left ideals of the coefficient ring. This correspondence is further explored here. All possibilities for the intersection of the ideals in a filter are catalogued, and the connections between filters and other ways of describing elementary radical classes are investigated. Some generalisations to nonassociative rings and groups are also presented.

Direct product of finite anti fuzzy normal sub-rings over non-associative rings

In this paper, we define the concept of direct product of finite anti fuzzy normal sub-rings over non-associative and noncommutative rings LA-rings and investigate the some fundamental properties of direct product of anti fuzzy normal sub-rings. © 2020 Global Research Online. All rights reserved.

A Kind of Variation Symmetry: Tarski Associative Groupoids (TA-Groupoids) and Tarski Associative Neutrosophic Extended Triplet Groupoids (TA-NET-Groupoids)

The associative law reflects symmetry of operation, and other various variation associative laws reflect some generalized symmetries. In this paper, based on numerous literature and related topics such as function equation, non-associative groupoid and non-associative ring, we have introduced a new concept of Tarski associative groupoid (or transposition associative groupoid (TA-groupoid)), presented extensive examples, obtained basic properties and structural characteristics, and discussed the relationships among few non-associative groupoids. Moreover, we proposed a new concept of Tarski associative neutrosophic extended triplet groupoid (TA-NET-groupoid) and analyzed related properties. Finally, the following important result is proved: every TA-NET-groupoid is a disjoint union of some groups which are its subgroups.

Fuzzy Ideals on Ordered AG-Groupoids

In this paper, we define the concept of direct product of finite fuzzy normal subrings over nonassociative and non-commutative rings (LA-ring) and investigate the some fundamental properties of direct product of fuzzy normal subrings.

The special linear group for nonassociative rings

Abstract We extend to arbitrary rings a definition of the octonion special linear group due to Baez. At the infinitesimal level, we get a Lie ring, which we describe over some large classes of rings, including all associative rings and all algebras over a field. As a corollary, we compute all the groups Baez defined.

Direct Product of Finite Intuitionistic Anti Fuzzy Normal Subrings over Non-associative Rings

Shal et. al cite:SKR, have introduced the concept of intuitionistic fuzzy normal subrings over a non-associative ring. In this paper, we investigate the concept of intuitionistic anti fuzzy normal subrings over non-associative rings and give some properties of such subrings

Literature Survey on Non-Associative Rings and Developments

In this paper we present a comprehensive survey and developments of existing literature of non-associative rings and enumerate some of their various applications in different directions to date. These applications explain the voluminous work in different fields of non-associative rings and through which various algebraic structures in theoretical point of view could be developed.

Hall's criterion for nilpotence in semi-abelian categories

A well-known theorem of P. Hall, usually called Hall's criterion for nilpotence, states: a group G is nilpotent whenever it has a normal subgroup N such that G/[N,N] and N are nilpotent. We widely generalize this result, replacing groups with objects in an abstract semi-abelian category satisfying suitable conditions. In particular, these conditions are satisfied in any algebraically coherent semi-abelian category, and hence in many categories of classical algebraic structures (including a few where Hall's theorem is already known). Note that the categories of crossed modules, crossed Lie algebras, n-cat groups, and cocommutative Hopf algebras over fields are also algebraically coherent. We give, however, a counter-example showing that Hall's theorem does not hold in the (semi-abelian) variety of non-associative rings.

Nonassociative rings satisfying a(bc) = b(ca) and (a, a, b) = (b, a, a)

In this paper, we study nonassociative rings satisfying the polynomial identities a(bc) = b(ca) and (a, a, b) = (b, a, a). We prove that nonassociative rings satisfying the identities a(bc) = b(ca) and (a, a, b) = (b, a, a) with characteristic are associative and commutative of degree five.

Ring theoretic aspects of quandles

We associate to every quandle X and an associative ring with unity k, a nonassociative ring k[X] following [3]. The basic properties of such rings are investigated. In particular, under the assumption that the inner automorphism group Inn(X) acts orbit 2-transitively on X, a complete description of right (or left) ideals is provided. The complete description of right ideals for the dihedral quandles Rn is given. It is also shown that if for two quandles X and Y the inner automorphism groups act 2-transitively and k[X] is isomorphic to k[Y], then the quandles are of the same partition type. However, we provide examples when the quandle rings k[X] and k[Y] are isomorphic, but the quandles X and Y are not isomorphic. These examples answer some open problems in [3].