Engineering Mathematics

Engineering Mathematics8 Aug 2026 · 17 min read

The Hopkins–Levitzki Theorem

Lam 4.14 and 4.15: semisimple equals J-semisimple plus left artinian, and the Hopkins-Levitzki theorem that over a semiprimary ring a module is noetherian if and only if artinia…

The Jacobson and Herstein Commutativity Theorems

The Jacobson and Herstein commutativity theorems (12.9) to (12.11), proved by the reduction ladder of (12.8) from division rings through primitive and semiprimitive rings, toget…

The Jacobson Density Theorem

The Jacobson–Chevalley Density Theorem: a semisimple module over a ring is acted on densely by that ring relative to its endomorphism division ring, with the Bourbaki proof, the…

The Jacobson Radical: Definition and Equivalent Characterisations

The Jacobson radical of a ring: its definition as an intersection of maximal left ideals, the annihilator and quasi-regularity characterisations, its left-right symmetry, and th…

The Krull–Schmidt Theorem

The Krull-Schmidt theorem for modules of finite length: existence of a decomposition into indecomposables from a single chain condition, uniqueness from the Krull-Schmidt-Azumay…

The Levitzki Radical and Locally Nilpotent Ideals

The Levitzki radical as the largest locally nilpotent ideal, Utumi's lemma on rings with ACC on right annihilators, Levitzki's theorem that nil one-sided ideals of a right noeth…

The Lower Nilradical (Baer Radical)

The lower nilradical of a ring: its definition as the radical of the zero ideal, its identification with the intersection of all prime ideals, why it is nil but not always nilpo…

Engineering Mathematics8 Aug 2026 · 17 min read

The Lower Nilradical of Polynomial and Matrix Rings

The Amitsur-McCoy theorem on the lower nilradical of a polynomial ring and its matrix analogue: polynomial and matrix rings are prime or semiprime exactly when the base ring is,…

Engineering Mathematics8 Aug 2026 · 22 min read

The Mal’cev–Neumann Construction of Laurent Series Rings

The Mal'cev-Neumann construction builds a division ring of formal series indexed by an ordered group with well-ordered supports. This page gives the construction, the key lemma …

The Multiplicative Group of a Division Ring

The multiplicative group of a division ring: its upper central series stabilises at the centre, so D star is nilpotent exactly when D is commutative. Includes the proof from the…

The Niven–Jacobson Theorem: Roots over Quaternion Algebras

The Niven-Jacobson theorem that quaternions over a real-closed field are right and left algebraically closed, with Niven's criteria for infinitely many roots and the exact count…

Engineering Mathematics8 Aug 2026 · 15 min read

The Opposite Ring and Left–Right Duality

The opposite ring construction and how it governs left-right duality: left modules as right modules over the opposite ring, transpose for matrix rings, anti-automorphisms, and w…

Engineering Mathematics8 Aug 2026 · 14 min read

The Radical and Quotients of Semilocal Rings

Lam (20.7): for a semilocal ring the radical of any quotient is the image of the radical, and quotients stay semilocal. Includes the general one-sided containment, why it can be…

The Radical of a Left Artinian Ring Is Nilpotent

Lam 4.12 and 4.13: in a left artinian ring the Jacobson radical is nilpotent and is the largest nilpotent one-sided ideal, so every nil left ideal is nilpotent. Covers the desce…

The Radical of a Module

The radical of a module: its definition as an intersection of maximal submodules, its identification with the sum of all small submodules, behaviour under submodules and direct …

Engineering Mathematics8 Aug 2026 · 17 min read

The Radical of a Quotient Ring

Lam's Proposition 4.6: when an ideal A lies inside the Jacobson radical of R, the radical of R/A is the image of rad R. Covers the correspondence proof, semiprimitivity of R mod…

The Radical of an Algebraic Algebra

For an algebra over a field, an element of the Jacobson radical is algebraic if and only if it is nilpotent. The consequence for algebraic algebras: the radical is the largest n…

Engineering Mathematics8 Aug 2026 · 13 min read

The Radical of an Ideal as an Intersection of Primes

The radical of an ideal in a noncommutative ring: the m-system definition, the theorem identifying it with the intersection of all primes above the ideal, why it is an ideal, an…

Engineering Mathematics8 Aug 2026 · 24 min read

The Radical of Matrix Rings and Power Series Rings

How the Jacobson radical behaves under two well-behaved ring constructions: full matrix rings, where it is computed entrywise, and formal power series rings, where it is the pre…

The Radical under Field Extension of Scalars

The Jacobson radical under scalar extension of a k-algebra to a field extension K: contraction and extension inclusions, nilness in the transcendental case, the trace-form argum…

The Radicals of a Ring Compared

A comparison of the prime, Levitzki, upper nil and Jacobson radicals of a ring: the chain of inclusions, the hypotheses that collapse it, explicit rings separating each pair, an…

Engineering Mathematics8 Aug 2026 · 24 min read

The Reduced Norm of a Cyclic Algebra

The reduced norm of a cyclic algebra, defined as the determinant of right multiplication on the algebra viewed as a K-space. Its polynomial version on the skew polynomial ring p…

The Socle of a Module and of a Ring

The socle of a module: the sum of its simple submodules, the largest semisimple submodule, the intersection of the essential submodules, the socle of a ring as a two-sided ideal…

The Structure of kG modulo Its Radical

Theorem (8.1) of Lam: the Wedderburn decomposition of kG modulo its radical, the isotypic decomposition of the regular module, the dimension formulas relating the simple modules…

The Unipotent Radical of a Linear Group

The unipotent radical of a linear group: existence and uniqueness of a largest normal unipotent subgroup, its characterisation by trivial action on composition factors and by th…

The Upper Nilradical and the Köthe Conjecture

The upper nilradical as the largest nil ideal of a ring, its position between the lower nilradical and the Jacobson radical, and Koethe's Conjecture together with its equivalent…

The Wedderburn–Artin Theorem: Structure of Semisimple Rings

The Wedderburn-Artin Theorem: every left semisimple ring is a finite product of matrix rings over division rings, with the endomorphism-ring proof, the uniqueness statement, the…

Theorem Dependency Map

A dependency map for noncommutative ring theory: the prerequisites and consumers of Wedderburn-Artin, Schur, the Density Theorem, Hopkins-Levitzki, Nakayama, Krull-Schmidt, Bass…

Triangular Rings: Ideals, Modules and Chain Conditions

The triangular ring built from two rings and a bimodule: its multiplication, the complete description of its left, right and two-sided ideals, its Jacobson radical, and the theo…

Engineering Mathematics8 Aug 2026 · 22 min read

Twisted Laurent Series Division Rings

Formal Laurent series with the twisted multiplication x a = sigma(a) x form a division ring. Computing its centre gives the first known centrally infinite division ring and, whe…

Unipotent Elements and the Lie–Kolchin–Suprunenko Theorem

Unipotent and lambda-potent linear transformations, the identification of unipotent elements with p-elements in characteristic p, and the Lie-Kolchin-Suprunenko triangularisatio…

Uniqueness in the Wedderburn–Artin Decomposition

Lam (3.8) and (3.9): decompositions of a ring into indecomposable two-sided ideals are unique as ideals, the isotypic sums of minimal left ideals are ideals that annihilate one …

Engineering Mathematics8 Aug 2026 · 23 min read

Units and Idempotents in Integral Group Rings

Character-theoretic arithmetic applied to integral group rings: central torsion units are trivial up to a root of unity, the centre of a finite group is determined by its integr…

Von Neumann Regular Rings

Von Neumann regular rings: the five equivalent conditions of Lam's theorem, left-right symmetry, the sandwich between semisimple and semiprimitive rings, the theorem that noethe…

Wedderburn’s Factorisation Theorem

Wedderburn's factorisation theorem: the minimal polynomial of an algebraic conjugacy class of a division ring splits into linear factors with roots drawn from that class, togeth…

Wedderburn’s Little Theorem: Finite Division Rings Are Fields

Wedderburn's Little Theorem: every finite division ring is a field. The class-equation proof with cyclotomic polynomials, the corollary that finite subrings of division rings ar…

Well-Ordered Subsets of Ordered Groups

Well-ordered subsets of a totally ordered set are characterised by the descending chain condition and by the existence of nondecreasing subsequences. In an ordered group they ar…

When Are eR and fR Isomorphic?

Two idempotents of a ring generate isomorphic principal right modules exactly when one is a product ab and the other is ba. This page proves that criterion, derives the induced …

When Is a Group Algebra a Local Ring?

When a group algebra is a local ring: the augmentation ideal of a p-group in characteristic p is nilpotent, the resulting equivalences for kG, the extension to group rings over …

Abelian Groups: Definitions, Properties and Examples

Abelian groups: axioms, standard examples, and why the commutative case is sufficient for computational number theory.

Engineering Mathematics7 Aug 2026 · 22 min read

Algebraic and Geometric Multiplicities of Eigenvalues

Algebraic against geometric multiplicity: degree of the characteristic polynomial, the bound on eigenspace dimension, eigenvalue counts and defective matrices.

Engineering Mathematics7 Aug 2026 · 2 min read

Algebraic Numbers and Minimal Polynomials

Algebraic numbers, minimal polynomials, algebraic integers, and the computational tests that distinguish them.

Algebras over a Ring

Algebras as rings carrying a compatible module structure, and the examples that matter computationally.

Algorithm Notation and Complexity Conventions

The notation, cost model and complexity conventions used throughout this collection, including the sub-exponential L-function.

Analysis of Berlekamp's Algorithm

Cost analysis of Berlekamp's algorithm and the variants that improve its dependence on field size.

Analysis of the Cantor-Zassenhaus Algorithm

The complete Cantor-Zassenhaus factorisation algorithm, its three stages, and its overall cost.

Engineering Mathematics7 Aug 2026 · 2 min read

Applications of the Hermite Normal Form

Solving integer linear systems, computing indices and intersections, and testing module membership using the Hermite normal form.

Approximating Functions by Random Sampling

Estimating a quantity by sampling, the sample size required, and the resulting confidence guarantee.