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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…
Content type
Guides
Structured, multi-part guides that take a subject from first principles to applied practice. 3043 pages.
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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…
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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…
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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…
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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…
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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…
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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…
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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…
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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…
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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…
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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…
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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…
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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 …
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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 …
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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 …
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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…
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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…
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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…
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Idempotents in I-Adically Complete Rings
Completion of a ring with respect to an ideal, the two conditions hidden in I-adic completeness, the implications from nilpotent to complete to contained in the radical, and the…
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Idempotents in Module-Finite Algebras over Complete Local Rings
Module-finite algebras over a complete commutative noetherian base: completeness passes to finitely generated modules, idempotents lift modulo the extended radical, endomorphism…
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Indecomposable Modules: First Examples and Pathologies
Indecomposable modules over finite-dimensional algebras: complete lists for two-dimensional algebras and for k[x] modulo x to the n, Lam's three-dimensional algebra with indecom…
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Indecomposable Rings and Connectedness
Indecomposable rings, centrally primitive idempotents, and the linkage relation on primitive idempotents. Blocks are the connected components of linkage, and over a right artini…
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Index of Named Theorems in Noncommutative Ring Theory
An index of the named theorems of noncommutative ring theory as numbered in Lam: Wedderburn-Artin, Schur, Hopkins-Levitzki, Nakayama, Maschke, Density, Krull-Schmidt-Azumaya, Ca…
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Integrality and the Degrees of Irreducible Characters
Integrality in the centre of a group algebra: the class sums are algebraic integers over the block idempotents, the irreducible degrees divide the group order, Schur's theorem s…
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J-Semisimplicity of Complex and Real Group Algebras
Rickart's theorem that the complex and real group algebras of an arbitrary group are Jacobson semisimple, presented as an algebraic positivity lemma plus an analytic lemma on th…
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Jacobson Rings, Finitely Generated Algebras and the Nullstellensatz
Hilbert rings, also called Jacobson rings, and the ascent of J-semisimplicity to finitely generated commutative algebras: Lam's results (5.3) to (5.5), Zariski's Lemma, and the …
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Jacobson Semisimple (Semiprimitive) Rings
Definition 4.7 and Proposition 4.8 from Lam: Jacobson semisimple or semiprimitive rings are those with zero Jacobson radical. Covers the definition, the simple modules and units…
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Left Primitive Rings That Are Not Right Primitive
One-sided primitivity: why left and right primitive are inequivalent, the comaximal left ideal criterion, Samuel's and Formanek's theorems on free algebras and semigroup rings, …
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Left–Right Symmetry: What Transfers and What Does Not
Which properties of a noncommutative ring are left-right symmetric and which are not: the opposite ring transfer principle, the symmetry of the Jacobson radical and semisimplici…
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Lifting Central Idempotents
When central idempotents lift from a quotient ring: the failure modulo the radical, the criterion of Lam (22.8), the correspondence modulo the square of a nilpotent ideal, Dade'…
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Lifting Idempotents Modulo an Ideal
Lifting idempotents from a quotient ring back to the ring: what liftability means, why an ideal inside the Jacobson radical makes isomorphism classes and primitivity descend, ho…
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Lifting Idempotents Modulo Nil Ideals
Lifting idempotents modulo a nil ideal: the binomial construction that turns an element idempotent mod I into a true idempotent in aR, uniqueness of the lift up to conjugacy, an…
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Linear Groups and the Burnside Problem
Linear groups as subgroups of GL(V) over a field: the General and Bounded Burnside Problems, the Trace Lemma bounding a matrix semigroup by the size of its trace set, and Burnsi…
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Local Rings and the Absence of Nontrivial Idempotents
Local rings characterised by a unique maximal one-sided ideal, by the non-units forming an ideal, and by a division residue ring; why a local ring has no nontrivial idempotents,…
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Local Rings: Definition and Equivalent Characterisations
Local rings in the noncommutative setting: the equivalence of unique maximal left ideal, unique maximal right ideal, division ring quotient by the radical, and closure of the no…
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m-Systems and the Characterisation of Prime Ideals
m-Systems in noncommutative rings: the definition, why multiplicatively closed sets are not enough, the characterisation of prime ideals as complements of m-systems, and the Zor…
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Maschke’s Theorem and Semisimplicity of Group Rings
Maschke's theorem in its modern two-sided form: kG is semisimple if and only if k is semisimple and the order of G is invertible in k. Includes the averaging proof, the auxiliar…
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Matrix Rings and Endomorphism Rings of Modules
Matrix rings and endomorphism rings of modules: the canonical isomorphism R = End(R_R), the identification End(M^n) = M_n(End M), matrix units, the centre and ideals of M_n(R), …
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Matrix Rings over Division Rings: The Model Simple Artinian Ring
Lam (3.3): matrix rings over division rings are simple, left and right artinian and noetherian, have a unique simple module given by the column space, and recover the division r…
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Matrix Units, Full Idempotents and Matrix Ring Recognition
Matrix units and full idempotents: how the corner at e11 in a matrix ring recovers the coefficient ring, the recognition theorem identifying a ring with a full set of matrix uni…
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Maximal Subfields of Division Rings
Maximal subfields of a division ring are characterised as self-centralizing subfields. Combined with the tensor product criteria this gives the perfect square dimension theorem,…
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Minimal Ideals and the Socle of a Primitive Ring
Minimal one-sided ideals and the socle of a primitive ring: Lam 11.9 to 11.11. The left-right transfer lemma for semiprime rings, why it fails without semiprimeness, and the the…
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Minimal Left Ideals in Semiprime Rings
Brauer's Lemma on minimal left ideals, the idempotent generation of minimal left ideals in a semiprime ring, and the theorem that a ring is semisimple exactly when it is semipri…
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Modules over Noncommutative Rings: Left, Right and Bimodules
Left modules, right modules and bimodules over a noncommutative ring: unital conventions, the identification of left R-modules with right modules over the opposite ring, endomor…
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Multiplicative Commutators in Division Rings
Multiplicative commutators in a division ring: the identity built from b equal to a minus one, the proposition that their common centralizer is the centre, the Cartan-Brauer-Hua…
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Nakayama’s Lemma in the Noncommutative Setting
Nakayama's Lemma for noncommutative rings: three equivalent conditions on a left ideal contained in the Jacobson radical, a full proof, the finite generation hypothesis and why …
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Nil and Nilpotent Ideals and the Radical
Lam 4.9 to 4.11: the definitions of nil and nilpotent one-sided ideals, the fact that a finite sum of nilpotent left ideals is nilpotent, and the proof that every nil one-sided …
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Noncommutative Ring Theory: A Structural Overview
An orientation to noncommutative ring theory as organised by Lam: semisimple rings and Wedderburn-Artin, the Jacobson radical as obstruction, density theory for primitive rings,…