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Key research themes
1. How do ring-theoretic properties of base rings extend to skew PBW and skew polynomial ring extensions?
This theme investigates the preservation and characterization of various algebraic properties—such as semicommutativity, symmetry, Baerness, and rigidity—when forming skew PBW extensions or skew polynomial rings over a given base ring. It is crucial for understanding how ring extensions behave structurally, impacting their ideal theory and module categories, which can influence applications like noncommutative algebraic geometry and coding theory.
Key finding: Introduces the concept of $Σ$-semicommutative rings relative to families of endomorphisms $Σ$ and proves that for such rings with compatible derivations, the Baer and quasi-Baer properties are equivalent and preserved under... Read more
Key finding: Extends results on symmetric properties from commutative rings and Ore extensions to the more general setting of skew PBW extensions. It demonstrates that under (Σ, Δ)-compatibility, weak symmetry of the base ring is... Read more
Key finding: Fully characterizes right projective ideals of skew polynomial rings R[x; σ] over hereditary Noetherian prime (HNP) rings, showing that any such projective ideal decomposes as Xb[x; σ], where X is invertible in the skew... Read more
Key finding: Studies generalized McCoy conditions in noncommutative rings, providing unifying frameworks that encompass various Armendariz-like and McCoy-like classes. It elucidates the transfer and preservation of weak McCoy conditions... Read more
2. What algebraic and module-theoretic structures are induced by skew polynomial rings and their polynomial evaluations?
This theme explores the structural description of modules over skew polynomial rings, the role of pseudo multilinear transformations (PMTs) in capturing module morphisms and polynomial evaluations, and how these constructions aid in understanding roots, ideals, and the center of skew polynomial rings. It bridges classical polynomial algebra with skew and multivariate extensions, broadening computational and theoretical tools within noncommutative algebra.
Key finding: Introduces and studies pseudo multilinear transformations (PMTs) as the key conceptual device linking modules over multivariate Ore extensions S = A[t; σ, δ] and their evaluation theory. The work proves a general product... Read more
Key finding: Develops a comprehensive Gröbner bases theory for graded two-sided ideals in skew polynomial rings S = P[s; σ], where σ is a monomial endomorphism compatible with ordering and divisibility, and embeds the free associative... Read more
Key finding: Studies algebraic identities involving skew Lie products combined with generalized derivations in prime rings with involution, yielding conditions that force commutativity and describe the structure of generalized... Read more
3. How do algebraic coding structures emerge from and interact with skew polynomial ring theory?
This theme investigates the applications of skew polynomial rings to coding theory, particularly the construction of skew constacyclic, skew quasi-cyclic, and skew BCH codes. The work reveals the unique algebraic features introduced by skew polynomial settings—noncommutativity, twisted partial actions, and automorphism-based constructions—and how these lead to new codes with improved parameters and decoding algorithms, bridging abstract ring theory with practical error correction.
Key finding: Surveys and advances the theory of skew Θ-λ-constacyclic codes via skew polynomial rings over finite fields and chain rings, detailing code structures, duality via Euclidean and Hermitian forms, and providing novel decoding... Read more
Key finding: Characterizes skew quasi-cyclic codes as left submodules of module rings over skew polynomial quotient rings, develops their generator and parity-check polynomials, and introduces similarity of polynomials to establish... Read more
Key finding: Defines and studies twisted partial skew power and Laurent series rings via group actions, examining primeness, semiprimality, ideal structure, and Goldie rank, providing conditions under which these properties hold. This... Read more