Guides

Notation and Symbols Reference

A notation reference for noncommutative ring theory: symbols for rings, ideals, radicals, modules, group rings and division algebras, with the variant conventions used by Jacobs…

Engineering Mathematics13 Aug 2026 · 21 min read

One-Sided Chain Conditions and the Dieudonné Example

Why the noetherian and artinian conditions are genuinely one-sided: the skew polynomial ring over a division ring with a non-surjective endomorphism, Dieudonne's finitely presen…

Open Problems: Köthe, Kaplansky and the Group Ring Conjectures

The open problems of noncommutative ring theory: Kothe's conjecture and its equivalent formulations, the unit, reduced, domain and semiprimitivity problems for group rings, and …

Engineering Mathematics13 Aug 2026 · 22 min read

Ordered Groups, Trivial Units and Domains

Ordered groups and their positive cones, the leading-term theorem making kG a domain with only trivial units, the complete answer to J-semisimplicity for abelian group algebras,…

Ordered Rings and Positive Cones

Ordered rings and positive cones: the three cone axioms, the dictionary between a compatible total order and its cone, the sign homomorphism, and the proof that an orderable rin…

Orderings on Division Rings

Orderings and preorderings specialised to division rings: every preordering is a normal subgroup of the multiplicative group containing all squares and all commutators, and the …

Perfect Rings and Semiprimary Rings

The definition of left and right perfect rings and of semiprimary rings, the implications between nilpotent, T-nilpotent and nil radicals, and the proof that semiprimary rings a…

Perfect Rings with Simple Quotient and Commutative Perfect Rings

Classification of perfect rings with simple semisimple quotient as matrix rings over local rings with T-nilpotent maximal ideal, and of commutative perfect rings as finite produ…

Polynomial Equations over Formally Real Division Rings

Corollary (18.12) of Albert's theorem: over a formally real division ring, a nonconstant central polynomial evaluated at an element has exactly the same centraliser as the eleme…

Polynomial Rings, Power Series and Laurent Series over a Ring

Polynomial, formal power series, Laurent polynomial and Laurent series rings over a possibly noncommutative ring: their construction, unit groups, degree and order functions, Ja…

Polynomials over Division Rings: Evaluation, Roots and Division

Polynomials over a division ring: why evaluation at an element is not a ring homomorphism, the definition of a right root, the noncommutative Remainder Theorem, and the conjugat…

Preorderings in Division Rings and the Intersection Theorem

The intersection theorem for preorderings of a division ring: division closure is automatic, the extension test for adjoining an element reduces to a single membership condition…

Preorderings in Rings

Preorderings in noncommutative rings: the two defining axioms, the permuted-product notation, the auxiliary preordering T sub b, and the theorem that a preordering is an orderin…

Prime and Semiprime Rings

Prime and semiprime rings defined by taking the zero ideal to be prime or semiprime: element-wise tests, the equivalence of semiprimeness with the absence of nonzero nilpotent l…

Prime Ideals in Noncommutative Rings

Prime ideals in noncommutative rings: the definition by products of ideals, the five equivalent characterisations of Lam (10.2) including the aRb element test, why maximal ideal…

Primitive Rings and Primitive Ideals

Left primitive rings and left primitive ideals in Lam 11.2 to 11.5: the definition via a faithful simple module, the failure of left-right symmetry, the identification of primit…

Primitive, Irreducible and Local Idempotents

Primitive, local and right irreducible idempotents: the three grades of indecomposability an idempotent can carry, how each is detected inside the corner ring eRe, the criterion…

Primitive, Simple, Prime and Semiprimitive: How the Classes Relate

How left primitive rings sit between simple and prime rings: Lam 11.6 to 11.8, the two-row implication chart, the collapse of all horizontal implications for left artinian rings…

Principal Indecomposable Modules

Principal indecomposable modules over a semiperfect ring: local idempotents, the simple top eR/eJ, the one-to-one correspondence with simple right modules, and the unique decomp…

Projective Covers

Projective covers: the definition via small kernels, the uniqueness theorem and its splitting argument, the standard existence examples over idempotents and T-nilpotent radicals…

Projective Covers over Semiperfect Rings

Construction of projective covers over semiperfect and right perfect rings: lifting a decomposition of M modulo the radical through local idempotents, the local-ring special cas…

Projective Modules over Local Rings Are Free

Why finitely generated projective modules over a local ring are free: the lifting lemma comparing projectives modulo an ideal inside the radical, the proof by Nakayama, invarian…

Projective Modules, Splitting and Semisimplicity

Projective modules: the lifting property, the equivalence with direct summands of free modules, and the homological characterisation of semisimple rings by projectivity or injec…

Radical Extensions and Kaplansky’s Theorem

Radical ring extensions require every element to have a power in the subring. For fields this forces characteristic p and one of two shapes, and for division rings radical over …

Reduced Rings as Subdirect Products of Domains

Lam's results (12.6) and (12.7): minimal primes of a reduced ring are completely prime, and a nonzero ring is reduced precisely when it is a subdirect product of domains, with t…

Right Algebraically Closed Division Rings

Baer's theorem and the classification of right algebraically closed division rings: a noncommutative centrally finite division ring in which every polynomial over the centre has…

Engineering Mathematics13 Aug 2026 · 15 min read

Right Perfect but Not Left Perfect: A Counterexample

The standard counterexample separating right perfect from left perfect rings: infinite strictly upper triangular matrices over a field, with proofs that the radical is right T-n…

Ring Class Hierarchy: Semisimple to Semilocal and Everything Between

A structural map of the main classes of noncommutative rings: the containment chain from semisimple down to semilocal, the parallel chain from division ring down to semiprime, a…

Ring Theory Conventions and Notation for Noncommutative Rings

Baseline conventions for noncommutative ring theory: rings with identity, subrings containing 1, two-sided ideals, simple rings, left and right zero-divisors, domains, units, De…

Rings of Stable Range One

Left stable range one: the definition, its relation to Bass' Theorem for semilocal rings, the cancellation theorem for modules whose endomorphism ring has stable range one, and …

Schur’s Lemma and Endomorphism Division Rings

Schur's Lemma: the endomorphism ring of a simple module is a division ring, homomorphisms between non-isomorphic simple modules vanish, and the sharper form over an algebraicall…

Schur’s Theorem on Torsion Linear Groups

Schur's theorem: a finitely generated torsion subgroup of GL(n,k) is finite over any field. The proof combines the bounded exponent lemma, the trace argument, and the abelian-by…

Semilocal Rings: Definition and Examples

Semilocal rings: the definition via a semisimple quotient by the Jacobson radical, the relation to having finitely many maximal ideals, closure under matrix rings and finite mod…

Semiperfect Endomorphism Rings and Module Decompositions

The endomorphism ring characterisation of semiperfect rings: End of a module is semiperfect exactly when the module is a finite direct sum of strongly indecomposable summands, w…

Semiperfect Rings and Decompositions of the Identity

The idempotent characterisation of semiperfect rings: primitive idempotents are local, the identity decomposes into finitely many orthogonal local idempotents, uniqueness up to …

Semiperfect Rings with Simple Quotient and Commutative Semiperfect Rings

Semiperfect rings with simple radical quotient are matrix rings over local rings, with n and k unique; commutative semiperfect rings are finite products of local rings; and the …

Semiperfect Rings: Definition and First Examples

Semiperfect rings: the definition as a semilocal ring in which idempotents lift modulo the Jacobson radical, why local and one-sided artinian rings qualify, matrix rings over lo…

Semiprime and Semiprimitive Rings as Subdirect Products

Lam's theorem (12.5): a nonzero ring is semiprime exactly when it is a subdirect product of prime rings, and semiprimitive exactly when it is a subdirect product of left primiti…

Engineering Mathematics13 Aug 2026 · 14 min read

Semiprime Ideals and n-Systems

Semiprime ideals in noncommutative rings: the squaring definition, the aRa element test, n-systems as complements, the lemma extracting an m-system from an n-system, and the the…

Engineering Mathematics13 Aug 2026 · 15 min read

Semiprimitive Rings and Faithful Semisimple Modules

Lam's Proposition 11.1: a ring is semiprimitive precisely when it carries a faithful semisimple left module. The proof, the explicit construction of that module from a complete …

Semisimple Rings: Definition and Equivalent Characterisations

Semisimple rings: the five equivalent conditions of Lam (2.5), the reduction of the whole notion to the regular module, the proof that semisimple rings are left artinian and lef…

Simple and Semisimple Modules: Definition and Equivalences

Simple and semisimple modules: the complement definition, the lemma that a nonzero semisimple module contains a simple submodule, and the theorem that semisimple, direct sum of …

Engineering Mathematics13 Aug 2026 · 25 min read

Simple Artinian Rings and Minimal One-Sided Ideals

Simple artinian rings: Lam's four-way equivalence between the DCC on left ideals, semisimplicity, the existence of a single minimal left ideal, and the matrix form over a divisi…

Engineering Mathematics13 Aug 2026 · 24 min read

Simplicity Criteria for Differential Polynomial Rings

Lam's criterion for the differential polynomial ring over a rational algebra to be simple: delta-simplicity of the base ring together with non-innerness of the derivation, with …

Engineering Mathematics13 Aug 2026 · 26 min read

Simplicity of Skew Laurent Polynomial Rings

Lam's theorem (3.18) following Jordan: three equivalent conditions for the skew Laurent polynomial ring to be simple, the inner order of an automorphism, full proofs of each imp…

Engineering Mathematics13 Aug 2026 · 22 min read

Skew Group Rings and Crossed Products

Skew group rings and crossed products: the multiplication rule, the associativity conditions on the action and the factor set, the identification of skew Laurent polynomial ring…