Engineering↗
Weak Boolean Products and Patchwork Properties
The relaxation of the Boolean product conditions that makes representations available more widely, and what is lost by the relaxation.
Content type
Guides
Structured, multi-part guides that take a subject from first principles to applied practice. 3043 pages.
Business↗
Wealth Planning for Australian Business Owners
Source fidelity note: This handbook preserves the supplied source's concepts while making their application explicit for practical business application and review.
Business↗
Website & Digital Presence
Source fidelity note: This handbook preserves the supplied source's concepts while making their application explicit for practical business application and review.
Business↗
What Is Artificial Intelligence? A Business Leader’s Guide
Understand artificial intelligence, narrow AI, machine learning capability and the practical questions leaders should ask before calling a system intelligent.
Engineering↗
What Universal Algebra Is: Scope and Method
Universal algebra studies what all algebraic structures have in common by stripping away the particular operations of groups, rings and lattices and asking which theorems surviv…
Business↗
Why Do You Need a Plan? A Tale of Two Designers
Source fidelity note: This handbook preserves the supplied source's concepts while making their application explicit for practical business application and review.
Engineering↗
Why Spring Design Creates Assembly Problems—and Cost
How spring geometry, clearance, end condition, solid height and handling can create assembly delay, rework and rejection.
Business↗
Win the Customer's Belief
Win the customer's belief by selling benefits, not features — translate every product feature into an impact on the customer's life, and focus on their success so they invest in…
Business↗
World-Class Customer Experience in 10 Points
A ten-point framework for world-class customer experience — listen, empathise, delight, keep customers updated, hire well, be honest, respond fast, always pick up, escalate clea…
Business↗
Zero Defect Manufacturing
Source fidelity note: This handbook preserves the supplied source's concepts while making their application explicit for practical business application and review.
Engineering↗
Absolutely Irreducible Modules and Schur’s Lemma over a Field
Absolutely irreducible modules over a k-algebra: the base-change lemma for Hom, the four equivalent characterisations in Lam (7.5), the role of Schur's Lemma and Burnside's Lemm…
Engineering↗
Additive Commutators in Division Rings
Additive commutators in a division ring: the centralizer of the set of all commutators is the centre, commutators generate D as a division algebra over its centre, and division …
Engineering↗
Amitsur’s Theorem on the Radical of a Polynomial Ring
Amitsur's Theorem on the Jacobson radical of a polynomial ring over an arbitrary ring: the radical is the coefficientwise extension of a nil ideal, proved by a roots-of-unity ar…
Engineering↗
Amitsur’s Theorem on the Radical of an Algebra of Small Dimension
Amitsur's theorem: for an algebra of dimension smaller than the cardinality of the base field, the Jacobson radical is the largest nil ideal. Full proof by linear dependence of …
Engineering↗
Archimedean Ordered Rings Are Commutative
Archimedean ordered rings: the equivalence of the archimedean property with the absence of infinitely large and infinitely small elements, and Hilbert's classification theorem s…
Engineering↗
Basic Idempotents
Basic idempotents in a semiperfect ring: the definition as an irredundant sum of primitive idempotents, the proof that such an idempotent is full, uniqueness of the corner ring …
Engineering↗
Basic Rings and Morita Reduction
Basic rings of semiperfect rings: the lattice and idempotent correspondences of Lam 25.8, the criterion that a semiperfect ring is basic precisely when its semisimple quotient i…
Engineering↗
Bass’s Theorem P: Characterisations of Perfect Rings
Bass's Theorem P characterises right perfect rings by descending chain conditions on principal left ideals, by DCC on cyclic submodules of left modules, and by an idempotent plu…
Engineering↗
Bass’s Theorem: Flat Modules Are Projective over Perfect Rings
Bass's theorem that flat right modules are projective exactly over right perfect rings, proved via projective covers and the flat module attached to a sequence of ring elements,…
Engineering↗
Behaviour of Simple Modules under Field Extension
How simple modules of a finite-dimensional algebra behave under scalar extension: every simple module over the extended algebra is a composition factor of an extended simple mod…
Engineering↗
Behaviour of the Radical under Ring Extensions
Lam's results (5.6) to (5.9) on how the Jacobson radical moves between a ring and an extension: direct summand and fixed-ring hypotheses for descent, finitely many centralising …
Engineering↗
Block Decomposition of a Semiperfect Ring
The block decomposition of a semiperfect ring: existence and uniqueness of the centrally primitive idempotents, the linkage relation on primitive idempotents, and why blocks of …
Engineering↗
Blocks of Finite-Dimensional Algebras and Group Algebras
Blocks of a finite-dimensional algebra over an algebraically closed field, described by central characters: the map from primitive idempotents to algebra homomorphisms from the …
Engineering↗
Burnside’s Theorem on Irreducible Subalgebras of Matrix Rings
Burnside's theorem: a subalgebra of End(V) acting irreducibly on a finite-dimensional vector space over an algebraically closed field is the whole endomorphism algebra. Proof by…
Engineering↗
Cancellation of Modules over Semilocal Rings
Lam (20.13): the four consequences of left stable range one for module theory — cancellation of finitely generated projectives, invariant basis number, freeness of stably free m…
Engineering↗
Catalogue of Counterexamples
A catalogue of the standard counterexamples in noncommutative ring theory: one-sided chain conditions, non-nil radicals, nil but not nilpotent ideals, failures of Krull-Schmidt,…
Engineering↗
Central Idempotents and Ring Direct Decompositions
Central idempotents and direct product decompositions of a ring: the correspondence with ideal direct sums, centrally primitive idempotents, uniqueness of the decomposition, and…
Engineering↗
Central Idempotents of a Group Algebra and Characters
Lam's Proposition 8.15: explicit formulas for the primitive central idempotents of a split semisimple group algebra in terms of irreducible characters, and for the class sums in…
Engineering↗
Centrally Finite and Centrally Infinite Division Rings
Centrally finite and centrally infinite division rings: why the centre of a division ring is a field, what finite dimension over the centre forces, the degree of a division alge…
Engineering↗
Chain Conditions: A Reference
A reference on chain conditions in noncommutative ring theory: noetherian and artinian modules and rings, the Hopkins-Levitzki theorem, descending chain conditions on principal …
Engineering↗
Chain Conditions: Noetherian and Artinian Rings and Modules
Ascending and descending chain conditions for modules and rings: equivalent formulations, the finitely generated criterion, behaviour in short exact sequences, composition serie…
Engineering↗
Characters of Modules over Finite-Dimensional Algebras
Characters of modules over a finite-dimensional algebra: additivity on short exact sequences, the characteristic zero theorem recovering composition factors, separation of absol…
Engineering↗
Completely Reducible Linear Groups
Completely reducible linear groups: the definition, the criterion that the spanned algebra be semisimple, inheritance by subnormal subgroups via Clifford's theorem, the finite i…
Engineering↗
Conjugacy Classes and Polynomials Vanishing on Them
Polynomials vanishing on a conjugacy class of a division ring: the degree bound of Lam (16.5), the description of the vanishing ideal as D[t] times the minimal polynomial (16.6)…
Engineering↗
Constructing Ordered Division Rings
How ordered division rings are built: the leading-coefficient ordering on a Mal'cev-Neumann Laurent series ring with well-ordered support, the hypothesis that the twist preserve…
Engineering↗
Corner Rings eRe and Their Radical
Corner rings eRe: the computation of their Jacobson radical as e times rad R times e, the injection of their left ideals and ideals into those of R, surjectivity for full idempo…
Engineering↗
Counting Irreducible Representations in Characteristic p
Brauer's count of modular irreducibles: the commutator subspace of a group ring, the basis of kG modulo the commutator subspace, the p-regular basis of kG modulo rad plus commut…
Engineering↗
Criteria for Central Finiteness and the Double Centralizer Theorem
The tensor product D over F with K opposite acts densely on D over its centralizer L. This yields equivalent criteria for finite dimension of a division subring, the double cent…
Engineering↗
Cyclic Algebras
The cyclic algebra construction of Dickson: generators, relations, dimension and simplicity, the identification of the centre and of K as a maximal subfield, and the norm criter…
Engineering↗
Dedekind-Finite Rings
Dedekind-finite rings: the definition, the module-theoretic reformulation, the proof that semilocal rings are Dedekind-finite, and Bass' Theorem that a coset a plus a left ideal…
Engineering↗
Deriving Wedderburn–Artin from the Density Theorem
The structure theorem for left primitive rings: every such ring is a dense ring of linear transformations over its endomorphism division ring, with the left artinian case recove…
Engineering↗
Differential Polynomial Rings and the Weyl Algebra
Differential polynomial rings k[x;delta]: how the Leibniz rule is forced by associativity, why inner derivations give nothing new, and how k_0[y][x;d/dy] is the first Weyl algeb…
Engineering↗
Division Rings and the Real Quaternions
Division rings and Hamilton's real quaternions: the norm and conjugation that produce inverses, the centre, the criterion for the analogous algebra over a general field, the Lip…
Engineering↗
Division Rings Containing an Algebraically Closed Field
If a noncommutative division ring contains an algebraically closed field of finite codimension, Artin-Schreier and Frobenius force it to be quaternions over a real closed centre…
Engineering↗
Division Rings: Basic Theory and Examples
Division rings from first principles: the multiplicative group, centre, centralizers, dimension transitivity, centrally finite versus infinite-dimensional examples, and the comm…
Engineering↗
Division-Closed Preorderings and the Artin–Schreier Method
The division closure of a preordering equals the intersection of all orderings containing it, the characterisation of preorderings that are intersections of orderings, and the c…
Engineering↗
Double Centralizers and Density for Bimodules
The bimodule form of density: centralisers and double centralisers, the equivalence of dense action with m-transitivity, the theorem that 2-transitivity forces the endomorphism …
Engineering↗
Embedding Free Rings in Division Rings
The free ring over a division ring embeds in a division ring. The proof places it inside the group ring of an orderable free group and applies the Mal'cev-Neumann series constru…