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…

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 …

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 …

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 …

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…

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…

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,…

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…

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…

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…

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 …

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…

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…

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…

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)…

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…

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 Mathematics8 Aug 2026 · 23 min read

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…

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 Mathematics8 Aug 2026 · 23 min read

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…

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…

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 Mathematics8 Aug 2026 · 22 min read

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…

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…

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…

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…

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…

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 …

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…

Examples of Noncommutative Local Rings

A catalogue of local rings from Lam section 19: localisations and germ rings on the commutative side, then division rings, twisted power series, constant-diagonal triangular mat…

Existence of Separable Maximal Subfields

A centrally finite division ring always possesses a maximal subfield separable over its centre, and every separable subfield extends to one. The proof combines the double centra…

Extending Orderings to Larger Rings

Extending orderings from a ring to a larger ring: the obstruction criterion, the Albert-Neumann-Fuchs theorem on unique extension to a ring of quotients, and the Gratzer-Schmidt…

FC Groups, the Δ-Subgroup and Trace Methods

FC groups and the finite conjugate subgroup Delta of a group: n-abelian groups via the transfer, torsion-free FC groups are abelian, Neumann's covering lemma, the truncation map…

Engineering Mathematics8 Aug 2026 · 22 min read

Fitting’s Lemma and Endomorphism Rings of Modules of Finite Length

Fitting's decomposition theorem for modules of finite length, the proof that indecomposable finite length modules have local endomorphism rings with nilpotent radical, and the e…

Flat Modules and Their Basic Theory

Flat modules: the definition through exactness of the tensor functor, the projective implies flat hierarchy, the torsion-free criterion over the integers, the exactness lemma fo…

Formally Real Division Rings

Formally real division rings: a twisted Laurent series family in which minus one is a sum of square-products but never a sum of squares, the Scharlau-Tschimmel level theorem, an…

Formally Real Rings and R. E. Johnson’s Theorem

Formally real rings: the weak preordering T(R) of sums of permuted doubled products, the definition of formal reality, R. E. Johnson's equivalence with orderability, and the div…

Free Rings and Rings Defined by Generators and Relations

Free rings over a coefficient ring, their universal property and monomial basis, and rings presented by generators and relations, including the Weyl algebra, generic zero-diviso…

Generalised Quaternion Algebras

Generalised quaternion algebras are the degree two cyclic algebras. This page gives their presentation, the standard involution and reduced norm, the proof that each is either a…

Glossary of Noncommutative Ring Theory

A glossary of noncommutative ring theory: precise definitions of semisimple, semiprimitive, primitive, prime, local, semiperfect and perfect rings, together with the module, ide…

Group Representations as Modules over the Group Algebra

How k-representations of a finite group correspond exactly to modules over the group algebra kG, why the correspondence is an isomorphism of categories, what it does to irreduci…

Group Rings and Semigroup Rings: Construction and First Properties

The group ring and semigroup ring construction over an arbitrary coefficient ring: the convolution product, the augmentation ideal, trivial and nontrivial units, zero-divisors f…

Group Rings in Prime Characteristic

Group rings over fields of prime characteristic: the Passman-Connell trace argument showing kG has no nonzero nil left ideals when G is a p-prime group, descent along algebraic …

Group Rings of Infinite Groups Are Never Semisimple

Lam's proposition that a group ring of an infinite group is never semisimple, with the finite-support proof, the sharper statement that the augmentation ideal is never a direct …

Herstein’s Lemma and Jacobson’s Commutativity Theorem

Herstein's Lemma on noncentral torsion elements of a division ring of characteristic p, its proof by eigenvectors of an inner derivation, and its three applications: Jacobson's …

Ideals of Matrix Rings and the Correspondence Theorem

Lam (3.1): the two-sided ideals of a full matrix ring are exactly the matrix rings over the ideals of the base, an inclusion-preserving lattice isomorphism, with the matrix-unit…

Idempotents and Direct Decompositions of Modules

How idempotents encode direct sum decompositions: the natural isomorphism Hom(eR, M) is Me, the ring isomorphism End(eR) is eRe, and the resulting characterisations of primitive…

Idempotents and the Peirce Decomposition

The Peirce decompositions attached to an idempotent: the two one-sided splittings of R into left and right ideals, the two-sided splitting into four additive summands, the corne…