Engineering articles and subject areas in the KEVOS knowledge library. 2176 pages.
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…
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: 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…
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…
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 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…
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 from first principles: the multiplicative group, centre, centralizers, dimension transitivity, centrally finite versus infinite-dimensional examples, and the comm…
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…
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 …
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…
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…
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 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 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…
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…
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: 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 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 are the degree two cyclic algebras. This page gives their presentation, the standard involution and reduced norm, the proof that each is either a…
A glossary of noncommutative ring theory: precise definitions of semisimple, semiprimitive, primitive, prime, local, semiperfect and perfect rings, together with the module, ide…
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…
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 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 …
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 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 …
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…
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…
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…
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…
Module-finite algebras over a complete commutative noetherian base: completeness passes to finitely generated modules, idempotents lift modulo the extended radical, endomorphism…
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…
Indecomposable rings, centrally primitive idempotents, and the linkage relation on primitive idempotents. Blocks are the connected components of linkage, and over a right artini…
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…
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…
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…
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 …
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…
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, …
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…
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'…
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…
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…
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…
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,…
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…
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…
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…