Engineering Mathematics articles in the KEVOS Engineering library. 1070 pages.
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…
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), …
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…
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…
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,…
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…
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…
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…
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…
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 …
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 …
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,…
Lam's results (8.4) to (8.8): normal p-subgroups act trivially on simple kG-modules, Clifford's theorem, the characterisation of the p-core by the radical, Wallace's formula for…
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…
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…
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 …
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: 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 and preorderings specialised to division rings: every preordering is a normal subgroup of the multiplicative group containing all squares and all commutators, and the …
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…
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…
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, 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 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…
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 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 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: the definition by products of ideals, the five equivalent characterisations of Lam (10.2) including the aRb element test, why maximal ideal…
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, 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…
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 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: the definition via small kernels, the uniqueness theorem and its splitting argument, the standard existence examples over idempotents and T-nilpotent radicals…
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…
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: the lifting property, the equivalence with direct summands of free modules, and the homological characterisation of semisimple rings by projectivity or injec…
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 …
Answers use KEVOS knowledge first and may use trusted web sources when needed. Verify critical decisions.