The theory of near-rings / / Robert Lockhart

• Record Nr.: UNINA9910508454403321
• Autore: Lockhart Robert (Mathematician)
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2021]; ©2021
• ISBN: 9783030817558; 9783030817541
• Descrizione fisica: 1 online resource (555 pages)
• Collana: Lecture Notes in Mathematics ; ; Volume 2295
• Disciplina: 512.4
• Soggetti: Near-rings; Anells associatius; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Intro -- 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 &gt -- 2) -- 6.4 Groups with Order pq Where p and q Are Prime and (p &lt -- q) -- 6.5 Groups with Order p2 -- 6.6 Groups with Order 2p2 (p &gt -- 2) -- 6.7 Groups with Order p3 (p &gt -- 2) -- 6.8 Groups with Order 2p3 or Order 2p4 (p &gt -- 2) -- 6.9 Groups with Order p4 (p &gt -- 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.

Contextual theology : skills and practices of liberating faith / / edited by Sigurd Bergmann and Mika Vahakangas

• Record Nr.: UNINA9910476837503321
• Pubbl/distr/stampa: 2021; London ; ; New York, New York : , : Routledge, , 2021; ©2021
• ISBN: 9780429348006; 0429348002; 9781000217261; 1000217264
• Descrizione fisica: 1 online resource (259 pages)
• Collana: Routledge New Critical Thinking in Religion, Theology and Biblical Studies
• Disciplina: 230.01; 230.09
• Soggetti: Theology - Methodology
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Includes index.
• Nota di contenuto: Foreword / Robert J. Schreiter -- Doing situated theology : introductory remarks about the history, method and diversity of contextual theology / Sigurd Bergmann & Mika Vähäkangas -- Can contextual theology bridge the divide? South Africa's politics of forgiveness as an example of a contextual public theology / Dion Forster -- Contextual theology on trial : African Pentecostalism, sacred authority, and sexual and gender based violence / Chammah Kaunda -- Gender, ethnicity and lived religion : challenges to contextual and liberation theologies / Elina Vuola -- Ecumenical liberation theology : how I experienced its arrival ; in Germany and Europe after 1968 / Ulrich Duchrow -- Economy, greed and liberation theology : a critique from a border location in India / Atola Longkumer -- The dissemination of Vikings : postcolonial contexts and economic meltdown / Sigríður Guðmarsdóttir -- Reclaiming tradition as critique of oppression / Teresa Callewaert -- Speaking from experience : comparing Mahdawi-Pentecostal approaches to equipment for mission and its theological justification / David Emmanuel Singh -- Theology in the Anthropocene--and beyond? / Sigurd Bergmann -- Theology of "eco-anxiety" as liberating contextual theology / Panu Pihkala -- Contextualization through the arts / Volker Küster -- World Christianity as post-colonializing of theology / Mika Vähäkangas.
• Sommario/riassunto: This book advances that history by exploring stories, images and discourses across a worldwide range of geographical, cultural and confessional contexts. Its twelve authors not only enrich our understanding of the significance of the contextual method, but also produce a new range of original ways of doing theology in contemporary situations. The authors discuss some prioritised thematic perspectives with an emphasis on liberating paths, and expand the ongoing discussion on the methodology of theology into new areas. Themes such as interreligious plurality, global capitalism, ecumenical liberation theology, eco-anxiety and the anthropocene, postcolonialism, gender, neo-pentecostalism, world theology, and reconciliation are examined in situated depth. Additionally, voices from Indigenous lands, Latin America, Asia, Africa, Australia, and Europe and North America enter into a dialogue on what it means to contextualise theology in an increasingly globalised and ever-changing world. Such a comprehensive discussion of new ways of thinking about and doing contextual theology will be of great use to scholars in Theology, Religious Studies, Cultural Studies, Political Science, Gender Studies, Environmental Humanities, and Global Studies.