Noncommutative Algebra and Ring Theory
Summary
Noncommutative algebra and ring theory explore algebraic structures in which the familiar commutative law ab = ba need not hold. At its core is the study of rings—sets equipped with two operations, addition and multiplication—and their modules, ideals and homological invariants. Beyond the classification of simple and semisimple rings, the field examines extensions such as Ore and PBW constructions, twisted polynomial and power series rings, and quantum algebras. Central topics include the behaviour of nilpotent and idempotent elements, duality and symmetry properties, and homological dimensions such as global and projective dimension. Interactions with representation theory, noncommutative geometry and mathematical physics have spurred the development of Calabi–Yau algebras, braided Hopf algebras and applications to mirror symmetry. Practical outcomes range from coding theory and cryptography to operator algebras in quantum mechanics and deformation theory. Common techniques involve lattice-theoretic approaches to ideal structure, categorical methods for module classification and computational tools for explicit examples.
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Noncommutative Algebra and Ring Theory publication trend
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Technical terms
Noncommutative ring: A ring whose multiplication need not satisfy ab = ba.
Ore extension: A noncommutative polynomial extension R[x; α, δ] defined by an endomorphism α and a derivation δ of R.
Armendariz ring: A ring in which the product of two zero‐annihilating polynomials forces each pair of coefficients to multiply to zero.
Skew generalized power series ring: A ring R[[S, ω]] constructed from R, a strictly ordered monoid S and a homomorphism ω: S → End(R).
Monoid: A set equipped with an associative binary operation and an identity element.
Skew PBW extension: A generalisation of noncommutative polynomial rings satisfying Poincaré–Birkhoff–Witt conditions for twisted variables.
References
- Classifications of Several Classes of Armendariz-like Rings Relative to an Abelian Monoid and Its Applications. Mathematics (2025).
- On Nilpotent Elements, Weak Symmetry and Related Properties of Skew Generalized Power Series Rings. Symmetry (2024).
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