Nearrings, Nearfields and Related Topics

The following sections are included:

• Dedication
• Note from the editors
• Contents

We are connected by our interest in the fascinating structures of near-rings (and related structures). Up to now, I attended all near-ring conferences (except the one in January 1985 at Nagarjuna University), so I am by far the most frequent participant at these conferences. But this time I regret that I cannot attend for health reasons. Let me wish all of you a most interesting conference; I am sure that the industrious and professional organizers, Professors Prasad, Srinivas, and Harikrishnan, will provide a fine environment for this conference.

It has long been known that the set M(G) of self-maps of an additive (but not necessarily Abelian) group G is a near-ring with respect to pointwise addition and composition of functions. Moreover, the set M₀(G) of zero-preserving self-maps of G is a zero-symmetric subnear-ring of M(G). Such nearrings and their subnear-rings have been extensively studied. Various authors have given this study a topological flavour by requiring that G be a topological group and considering the sets N(G) and N₀(G) of continuous and continuous, zero-preserving self maps of G, respectively. We will survey the main results of this study, and also some more recent ones concerning certain subnear-rings of N(G) and N₀(G).

In this article, we review the development of planar nearrings in the past ten years.

The following sections are included:

• Introduction
• Elementary concepts in vector spaces and modules over rings
• Finite Goldie dimension in modules
• Finite spanning dimension in modules
• Goldie dimension in N-Groups (where N is a near-ring)
• Finite spanning dimension in N-groups
• Bibliography

We introduce different prime radicals for near-rings and near-ring modules.

The following sections are included:

• Introduction
• Polya’s Recurrence Theorem
• Burnside’s Problem
• Conclusion
• Bibliography

Composition p-rings and composition t-rings are two types of composition rings introduced to facilitate the study of composition rings; in particular those that have foundation zero.

This chapter introduces background, history, and recent results on centers and generalized centers of near-rings.

The following sections are included:

• Introduction
• Categories
• Normal Categories from Nearrings
• Bibliography

Historically ideal theory for lattices was developed by Hashimoto. He established that there is a one to one correspondence between ideals and congruence relations of a lattice L under which the ideal corresponding to a congruence relation is a whole congruence class under it if and only if L is a generalized Boolean algebra. His proof involved topological ideas which were later simplified, using lattice theoretic ideas, by Gratzer and Schmidt. Also Gratzer introduced the notion of standard elements and ideals in lattices which were extensively studied by Gratzer and Schmidt. It was shown that standard ideals of lattices play a role somewhat similar to that of normal subgroups of groups or ideals of rings. Later, Fried and Schmidt extended the notion of standard ideals of lattices to convex sublattices. Generalizations of some of these results to trellises (or also called weakly associative lattices) may be found in Bhatta and Ramananda and Shashirekha.

The concept of a hypernearring is a generalization of the concept of a nearring. The overall aim of this paper is to present an introduction to some of the results, methods and ideas about hypernearrings. We study:

• (1) Basic concepts of algebraic hyperstructures and hypernearrings;
• (2) Fundamental relation on hypernearrings;
• (3) Hyper R-subgroups of a hypernearring;
• (4) H_v-nearrings.

The theory of fuzzy set which was introduced by Zadeh is applied to many area of mathematics. In this paper, we review some results of fuzzy ideals in a nearring and then we generalize these results to hypernearrings. We study:

• (1) Fuzzy subnearrings and ideals of a nearring;
• (2) Hypernearrings;
• (3) Fuzzy sets and hyperideals;
• (4) Fuzzy hyper R-subgroups;
• (5) Characterizations of sub-hypernearrings by using triangular norms.

In this paper, we introduce weakly automorphism N-invariant module, which is a generalization of both the concepts weakly N-injective and automorphism N-invariant. We obtain some basic properties and results of such modules. Also we characterize semisimple ring, CEP-ring and injective module by properties of weakly automorphism N-invariant module.

Recall that an n-by-n generalized matrix ring R is defined in terms of sets of rings (R_i, R_j)- bimodules {M_ij}, and bimodule homomorphisms θ_ijk : M_ij ⊗_Rj M_jk → M_ik, where the set of diagonal matrix units {E_ii} form a complete set of orthogonal idempotents. Moreover, an arbitrary ring R with a complete set of orthogonal idempotents has a Peirce decomposition which can be arranged into an n-by-n generalized matrix ring R^π which is isomorphic to R. In this paper, we focus on the subclass T_n of n-by-n generalized matrix rings with θ_iji = 0 for i ≠ j. T_n contains all upper and all lower generalized triangular matrix rings and is called the class of n-by-n trivial generalized matrix rings. This paper is primarily an expository paper based on a plenary talk presented at the 24th International Conference on Nearrings, Nearfields, and Related Topics. However some new results are presented at the end of the paper.

The following sections are included:

• Preliminary remarks
• Introduction
• Some Basic Results
• Modules over matrix nearrings and primitivity
• Left and two-sided ideals
• Conclusive remarks
• Bibliography

The following sections are included:

• Preliminaries
• Boolean near-rings
• One sided ideals in Boolean near-rings
• Subdirect product representation of Boolean near-rings
• Radical of a zero-symmetric Boolean near-ring
• Special Boolean near-rings
• Bibliography

Let R be a commutative ring with nonzero identity, Z(R) be its set of zero-divisors, and if a ∊ Z(R), then let ann_R(a) = {d ∊ R | da = 0}. The annihilator graph of R is the (undirected) graph AG(R) with vertices Z(R)* = Z(R) \ {0}, and two distinct vertices x and y are adjacent if and only if ann_R(xy) ≠ ann_R(x)∪ann_R(y). It follows that each edge (path) of the zero-divisor graph Γ(R) is an edge (path) of AG(R). The extended zero-divisor graph of R is the undirected (simple) graph EG(R) with the vertex set Z(R)*, and two distinct vertices x and y are adjacent if and only if either Rx∩ann_R(y) ≠ {0} or Ry∩ann_R(x) ≠ {0}. Hence it follows that the zero-divisor graph Γ(R) is a subgraph of EG(R). In this paper, we collect some properties (many are recent) of the two graphs AG(R) and EG(R).

Let I be an ideal of a nearring N. We introduce the notions of equiprime graph of N denoted by EQ_I (N) and c-prime graph of N denoted by C_I (N). We relate EQ_I (N), C_I (N) and the graph of a nearring with respect to an ideal, G_I (N). We prove that diam(EQ_I (N\I)) ≤ 3 and diam(C_I (N\I)) ≤ 3 and deduce that the prime graphs are edge partitionable. It is well-known that the homomorphic image of a prime ideal need not be a prime ideal in general. We study graph homomorphisms and obtain conditions under which the primeness property of an ideal is preserved under nearring homomorphisms.

The following sections are included:

• Introduction
• Probabilistic paranormed Spaces
• Copulæ
• Bibliography

We look at structure results in nearring theory, especially with regard to Jacobson style primitivity. We look at some applications in state automata and cellular automata, where we find interesting structures. We then look at generalising planarity and nearfields, some of the best structured nearrings, and find that Jacobson style primitivity plays an important role.

The paper explores some of our recent work and presents several ways forward that we think might be of value. While we hope to answer some of the questions that we raise here, it is likely that other researchers will find simpler ways forward that we are unable to see. We look forward to some surprises!

We show that a Lie ring P over C(X), the ring of real valued continuous functions on X, is isomorphic to a Lie ring of the set of all sections of a Lie algebra bundle ξ denoted by Γ(ξ), if and only if P is finitely generated projective module over C(X), where X is a compact Hausdorff space. The above results are extended to arbitrary topological space X for bundles of finite type. This result gives rise to a common intuition that a study of rings which are also projective modules over commutative rings are like the study of algebra bundles over Topological spaces.

The following sections are included:

• A Brief Introduction
• Some Important Results
• Bibliography

In 1933, P. Jordan proposed a quantum mechanical formalism in which the operators form only a near-algebra. In this regard, the investigation of near-algebras is interesting not only as an axiomatic question but also for physical reasons. A near-algebra is a near-ring which admits a field as a right operator domain. Near-algebras occur more naturally as mappings of linear spaces. The concept Gamma ring (or Γ-ring) (a generalization of a ring) was introduced by Nobusawa [5] and generalized by Barnes [1]. A generalization of both the concepts near-ring and the Γ-ring namely Γ-near-ring was introduced by Satyanarayana [6]. Brown [2], Yamamuro [11], Irish [3] and Srinivas [4;7;8;9] have studied the structure of near-algebras. Γ-near-algebra is a generalization of both the concepts of nearalgebra and Γ-near-ring is introduced by Srinivas et al. [10]. Throughout this lecture, “near-algebra” we mean a “right near-algebra”, and X denotes a field.

The following sections are included:

• What is the n–gen problem?
• Previous results
• The size of the problem
• The complexity of the problem
• Bibliography

Permutation Polynomials over finite rings have many applications, including cryptography. The behaviour of bivariate permutation polynomials that form Latin squares over some finite rings is studied. The linear bivariate permutation polynomials for all finite rings and quadratic bivariate permutation polynomials for all rings of even order have been characterized. The Latin squares formed by some permutation polynomials are considered over a few small rings and studied with regard to few Latin square properties.

The following sections are included:

• List of participants
• Conference tour
• Nearring conferences