The Theory of Near-Rings

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 < q)
6.5 Groups with Order p2
6.6 Groups with Order 2p2 (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