The Theory of Near-Rings

Description

This book offers an original account of the theory of near-rings, with a considerable amount of material which has not previously been available in book form, some of it completely new. The book begins with an introduction to the subject and goes on to consider the theory of near-fields, transformation near-rings and near-rings hosted by a group. The bulk of the chapter on near-fields has not previously been available in English. The transformation near-rings chapters considerably augment existing knowledge and the chapters on product hosting are essentially new. Other chapters contain original material on new classes of near-rings and non-abelian group cohomology. The Theory of Near-Rings will be of interest to researchers in the subject and, more broadly, ring and representation theorists. The presentation is elementary and self-contained, with the necessary background in group and ring theory available in standard references. Foreword by Günter Pilz Preface Notation Gothic Symbols Contents Part I Structure Theory 1 Stems, Mappings and Near-Rings 1.1 Basic Group Theory 1.1.1 Sylow Theory 1.1.2 The Jordan-Hölder Theorem 1.1.3 Solvable, Supersolvable and Nilpotent Groups 1.2 Homological Algebra and Category Theory 1.3 Topology 1.3.1 The Kuratowski Closure Axioms 1.4 Stems and Near-Rings 1.4.1 Star Notation 1.4.2 Pre-Near-Rings 1.4.3 Conventions and Notation 1.4.4 Examples of p.n.r. and of Near-Rings 1.5 Hosting 1.6 Ideals 1.7 Subdirect Products of Near-Rings 1.8 Sideals and Near-Ring Groups 1.8.1 Generalisations 1.8.2 Right Near-Ring Groups 1.8.3 Highly Non-standard Terminology 1.8.4 Sub-Structures and Mideals 1.8.5 Faithfulness 1.8.6 Monogenicity 1.8.7 Some Two-Sided Sideals 1.8.8 The Weak Left Ideal Property 1.8.9 Rings and Modules 1.9 Semi-simplicity 1.10 Prime and Semi-prime Ideals 1.10.1 Prime Ideals 1.10.2 Semi-prime Ideals 1.10.3 Complements of Prime and Semi-prime Ideals 1.10.4 Prime and Semi-prime Ideals 1.11 Near-Fields 1.12 A-Matrices 1.13 Functions and Function Composition 1.14 The δ Operator and Phomomorphisms 1.14.1 The δ Operator 1.14.2 Phomomorphisms 1.15 Annihilators 1.16 Conjugacy and Annihilators 1.17 Sylow Subgroups 1.18 The Zeroiser Ideal 1.19 The Core of a Left Ideal 1.20 Anti-chains of Subgroups 1.21 Subsets 1.21.1 Generating Near-Rings 1.21.2 Lifting Near-Rings 1.22 Nil and Nilpotent Sets 1.22.1 Sums of Nil Ideals 1.22.2 Sums of Nilpotent Ideals 1.23 Cores 1.24 Classes of Near-Rings 1.24.1 Distributively Generated and F-Near-Rings 1.24.2 Class F Near-Rings 1.24.3 The Fj Cores, (j = 1,2, 3) 1.24.4 Constant and Near-Constant Near-Rings 1.24.5 Opposites 1.24.6 Non-Zero-Symmetric Near-Rings 1.25 The Distributor and the Annular Ideal 1.25.1 The Distributor Ideal 1.25.2 The Multiplicative Centre 1.25.3 The Annular Ideal 1.26 Bi-distributive Stems 1.27 Subgroup Series 1.27.1 Weak Distributivity 1.27.2 Annularity 1.27.3 N(+)-Nilpotence 1.28 Modular Ideals 1.29 Quasi-regular Left Ideals 1.29.1 Quasi-regularity in Rings 1.30 Pseudo-Rings 1.31 Propriety 1.31.1 ``Left'' and ``Right'' Confusion 1.31.2 Proper Structures 1.31.3 Transferred Epithets 1.31.4 Problematic Terminology 1.32 An Unsettling Homomorphism 2 Near-Ring Theory 2.1 Pre-Near-Ring Construction Conditions and the Associativity Core 2.1.1 Host Determination Strategies 2.1.2 Distributive Generation 2.1.3 Co-structures: A Sort of Duality 2.1.4 Reduced Free Groups and Another Sort of Duality 2.1.5 Construction Conditions and F-Near-Rings 2.1.6 Bounds on Associativity Checking 2.2 Coupling and Dickson Near-Rings 2.2.1 D-Near-Rings 2.3 Affine Near-Rings 2.4 Near-Rings Hosted by Semi-direct Products 2.4.1 Near-Rings Hosted by Dn 2.5 Ideas from Mathematical Logic and Universal Algebra 2.5.1 Equational Products 2.5.2 Boolean Algebras and Boolean Rings 2.5.3 Boolean Near-Rings 2.5.4 Finite Boolean Near-Rings 2.5.5 Partially Ordered Sets 2.5.6 Lattices 2.5.7 Finiteness Conditions: Chains, Intersections, Generators 2.5.8 Ultra-Products 2.6 Adjoining an Identity 2.7 Planarity 2.7.1 The Ferrero Construction 3 Near-Fields 3.1 Near-Fields 3.1.1 Near-Fields Not of Characteristic 2 3.1.2 General Near-Fields 3.2 Commutators and the Sub-near-Field L 3.2.1 The Sub-near-Field F 3.3 Finite Near-Fields 3.3.1 The Smallest Proper Near-Field 3.3.2 General Cases 3.3.3 The Normal Core of D* 3.3.4 The Multiplicative Centre 3.3.5 The Multiplicative Group Structure of Finite Near-Fields 3.3.6 Presentations for Finite Near-Fields with S2 Cyclic 3.3.7 Z-Group Properties 3.3.8 The Product of All the Non-zero Elements 3.4 Finite Dickson Near-Fields 3.4.1 Coupling Maps and Dickson Near-Fields 3.4.2 A Theorem Reported by Marshall Hall 3.4.3 The Smallest Proper Near-Field Having All Sylow Subgroups Cyclic 3.4.4 The Algebra of the Dickson Process 3.4.5 A Generalisation of the Dickson Process 3.4.6 The Historical Dickson Process 3.4.7 When N* Is a Z-Group 3.4.8 Multiplication in Finite Dickson Near-Fields 3.4.9 Isomorphism in Finite Dickson Near-Fields 3.4.10 Sub-near-Fields 3.4.11 Number-Theoretic Issues 3.4.12 Near-Field Automorphisms 3.4.13 Prime Divisors of δ: Hall's Theorem 3.4.14 L and N 3.4.15 An Intrinsic Characterisation of Dickson Near-Fields 3.5 Group Structure of N* 3.5.1 Presentations for Solvable Near-Fields with S2 Quaternionic 3.5.2 Presentation for Non-Dickson Solvable Cases 3.6 Frobenius Groups 3.6.1 Basics 3.6.2 Sharply 2-Transitive Groups 3.6.3 Affine Groups 3.6.4 Near-Fields to Sharply 2-Transitive Groups 3.6.5 Further Affine Groups 3.6.6 Sharply 2-Transitive Groups to Near-Fields 3.6.7 Dickson and Non-Dickson Near-Fields 3.7 Finite Non-Dickson Near-Fields 3.7.1 A Classification Lemma 3.7.2 Element Orders 3.8 General Finite Non-fields 3.9 Infinite Near-Fields 3.9.1 Characteristic Zero 3.10 A Continuing Story Part II Near-Rings Hosted by Classes of Groups 4 Near-Rings on Groups with Low Order 4.1 Small Non-abelian Groups 4.1.1 Groups with Order 16 4.1.2 Groups with Order 18 4.1.3 Non-abelian Groups with Order 21 4.1.4 Groups with Order 24 4.1.5 Groups with Order 27 4.1.6 Coda 5 Near-Rings on Some Families of Groups 5.1 Finite Symmetric Groups 5.2 Finite Simple Non-abelian Groups 5.2.1 Isotopy 5.2.2 A Class of Non-trivial Near-Rings Hosted by Any Group 5.3 Unital Near-Rings on Sn 5.4 The Quaternion Group with Order 8 5.4.1 Unital d.g. p.n.r. Hosted by Q8 5.5 Dihedral Groups 5.5.1 The Dihedral Group of Order 8 5.5.2 Other Finite Dihedral Groups 5.5.3 Pre-Near-Rings 5.5.4 The Infinite Dihedral Group 5.6 Finite Groups from the Krimmel Class 5.6.1 A Classification Theorem Reported in Gorenstein 5.7 Generalised Quaternion Groups 5.8 Dicyclic Groups 5.9 Finite Hamiltonian Groups 5.10 Semi-dihedral Groups 5.11 Gorenstein's Group Mm(p) 5.12 Central Products 5.13 Free Products 5.14 Finite Non-solvable Groups 5.14.1 Groups with Order 360 5.14.2 Groups with Order 600 5.14.3 Groups with Order 720 5.14.4 Remaining Possibilities with Order 720 5.14.5 Direct Sums of Simple Groups 6 Near-Rings Hosted by p-Groups and Related Groups 6.1 Groups with Order p 6.2 The Klein Group 6.3 Groups with Order 2p (p > 2) 6.4 Groups with Order pq Where p and q Are Prime and (p 2) 6.7 Groups with Order p3 (p > 2) 6.8 Groups with Order 2p3 or Order 2p4 (p > 2) 6.9 Groups with Order p4 (p > 2) 6.10 The Prüfer Groups 6.11 A Research Suggestion Part III Representations and Cohomology 7 Transformation Near-Rings 7.1 Introduction 7.2 Preliminaries 7.2.1 Mapping Notation 7.2.2 Ideals of T(N) 7.2.3 Automorphisms of T(N) 7.2.4 The Finite Topology 7.2.5 Sub-near-Rings 7.2.6 E(N), I(N), A(N), B(N), and Phom(N) 7.3 Multiplicative Structure 7.3.1 Sideals and Cleiks 7.3.2 A-Matrices 7.3.3 Operating on (a,b) 7.3.4 Left and Right Sideals 7.3.5 Nilpotence 7.3.6 Idempotence 7.3.7 T0(N) Generalised 7.3.8 A Sub-near-Ring of T0(S3) 7.4 T(N), H(N) and B(N) 7.4.1 The Structure of H(N) 7.4.2 The Structure of T(N) 7.4.3 More on the Representation 7.4.4 Permutations and Additive Isomorphisms 7.4.5 Automorphisms of T0(N) 7.4.6 The Structure of B(N) 7.4.7 Further Investigation 7.5 Some Examples 7.5.1 The Cyclic Group C3 7.5.2 Finite Dihedral Groups 7.5.3 Dn when n Is Odd 7.5.4 Dn when n Is Even 7.5.5 D∞ and A(D∞) 7.5.6 Q8 7.6 Additive Structure 7.6.1 M(N) 7.6.2 Centraliser Near-Rings 7.6.3 A Duality of Semi-Groups 7.6.4 Density 7.7 MS() when S Is Fixed-Point-Free 7.7.1 The Structure of Minimal Left Ideals 7.7.2 Right Near-Ring Groups 7.7.3 Annihilators 7.7.4 Chains of Left Ideals 7.7.5 Simple Near-Rings 7.7.6 Left Ideals 7.7.7 Modular Left Ideals 8 Generalisations and Sub-near-Rings of Transformation Near-Rings 8.1 Commutators 8.2 More Sub-near-Rings 8.2.1 Special Cases 8.3 Hadamard Products 8.4 Endomorphism Near-Rings 8.4.1 Related Sub-near-Rings 8.4.2 Sequences of Endomorphism Near-Rings 8.5 Other Kinds of Endomorphism Near-Ring 8.6 Change of Groups 8.6.1 Near-Loops 8.6.2 Homomorphisms and Normal Sub-Loops 8.6.3 The Host Problem 8.6.4 Transformations on Near-Loops 8.6.5 Transformations on Sets 8.7 The Stemhome Near-Ring 8.7.1 The Stemhome Functor 8.8 The Wurzel 8.9 Elementary Closure Procedures 8.9.1 Additive and Multiplicative Closures 8.9.2 A Topological Closure 8.10 Polynomials 8.10.1 Near-Rings 8.10.2 Skew Polynomial Near-Rings 9 Phomomorphisms 9.1 General Theory 9.1.1 Extending Mappings to Phomomorphisms 9.1.2 Phomomorphism-Invariant Subgroups 9.2 Cohomology Groups 9.2.1 Non-abelian Group Cohomology 9.2.2 Cochain Symmetries 10 Specific Examples 10.1 The Group B2(Z3,Q8) 10.2 Phom (Q8) 10.3 Phom (S3) 10.4 Phom (D4) 10.5 Phom (Z2,[A4,A4]) Part IV Some Traditional Constructions 11 Modules 11.1 Introduction 11.2 Basics 11.2.1 Sub-structures 11.3 Primitivity: Laxton's Paper 11.4 Primitive Near-Rings 11.4.1 I(N), A(N) and E(N) 11.4.2 The Density Theorem 11.4.3 Unital, 2-Primitive Non-rings 11.4.4 Left Ideal Cores 11.4.5 A Famous Result of Fröhlich 11.4.6 Near-Fields 11.4.7 MS() 11.4.8 Alternative Generalisations 11.5 Module-Like Structures 11.6 A-Matrices 11.6.1 Left Near-Ring Groups 11.6.2 Right Near-Ring Groups 11.6.3 Opposites 11.6.4 Bimodals 11.6.5 Kronecker Products 11.6.6 Applications to Bimodal Near-Ring Groups 11.6.7 Multiple Near-Ring Groups 12 Radicals 12.1 Abstract Radicals 12.1.1 General Radicals 12.1.2 Divinsky's Definition 12.1.3 Hoehnke Radicals 12.2 Rings and Modules 12.2.1 The Köthe Radical 12.2.2 The Jacobson Radical for Rings 12.2.3 Chain Conditions in Rings 12.3 Near-Rings 12.3.1 Radical Maps 12.3.2 Near-Ring Radicals 12.3.3 Chain Conditions in Near-Rings 12.3.4 The Ring Radical 12.3.5 The Nil Radical 12.3.6 The Prime Radical 12.3.7 Prime Ideals in Ring Theory 12.3.8 Further Work 12.4 The Trouble with Generalisations 13 Matrices 13.1 The Work of Meldrum and van der Walt 14 F-Near-Rings 14.1 The Pseudo-Centre 14.2 Ideals 14.3 F-Stems Part V Product Theory 15 Product Theory 15.1 General Case 15.2 The Stem (N, +, ) 15.3 Additive and Multiplicative Splitting 15.4 A Particular Situation 15.4.1 The Stemhome 15.4.2 Associative and Distributive Elements 15.4.3 Opposites 15.4.4 Coupling 15.4.5 Further Symmetries 15.4.6 Isomorphisms 15.4.7 Isotopies 15.5 Restricted Co-domains 15.5.1 Bi-F-Stems 15.6 Abelian Groups 15.7 General Groups 16 Product Theory on Finite Elementary Abelian Groups 16.1 Finite Elementary Abelian p-Groups 16.1.1 Matrix Representations 16.1.2 The Cayley-Hamilton Theorem 16.1.3 Restricted Transposes 16.1.4 Notational Laxity 16.1.5 Identities 16.1.6 The Klein Group 16.1.7 Subfields 16.2 Finite Near-Fields 16.3 Near-Fields Hosted by (Cp Cp) 16.3.1 Non-central Cases (the Fantastic Four) 16.4 The Near-Field with Order 9 16.5 Finite Division Rings 16.6 Near-Fields Hosted by (Cp Cp) with p > 3 16.7 The Fantastic Four 16.7.1 The Condition πs = πrs 16.7.2 The Subgroup M* 16.7.3 Determinants for the Fantastic Four 16.8 Eigenvalues 16.9 Apologia A Isotopy A.1 Introduction A.1.1 Additive Conjoints and Near-Rings A.1.2 Multiplicative Conjoints and Opposites A.1.3 Isotopy A.1.4 Isomorphs Which Are Not Allotropes A.1.5 Basic Results in Isotopy A.1.6 Alleles A.2 A-Matrices A.2.1 Matrix Symmetries A.2.2 Listing Classes of Products A.2.3 Associativity B Near-Ring Products on D4 B.1 Seven Non-isomorphic Left Near-Rings Hosted by D4 B.2 Additive Conjoints C Other Structures of Interest C.1 Some Related Algebras C.1.1 Semi-Rings C.1.2 Planar Ternary Rings C.1.3 Alternative Division Rings C.1.4 Semi-Fields C.1.5 Neo-fields C.1.6 Quasi-Fields C.1.7 Quasi-Groups and Loops C.1.8 Near-Domains C.1.9 L-R Systems C.1.10 Other Structures and Structural Coding C.2 Some Geometry C.2.1 Projective Spaces C.2.2 Finite Projective Planes D Semi-Linear Mappings E Zsigmondy's Theorem Afterword Bibliography Index