小 Krull 维数 Ore 扩张的性质 $(\diamond)$

Isomorphism problems and groups of automorphisms for Ore extensions $K[x][y; f\frac{d}{dx} ]$ (prime characteristic)

Let $Λ(f) = K[x][y; f\frac{d}{dx} ]$ be an Ore extension of a polynomial algebra $K[x]$ over an arbitrary field $K$ of characteristic $p>0$ where $f\in K[x]$. For each polynomial $f$, the automorphism group of the algebras $Λ(f)$ is explicitly described. The automorphism group ${\rm Aut}_K(Ł(f))=S\rtimes G_f$ is a semidirect product of two explicit groups where $G_f$ is the {\em eigengroup} of the polynomial $f$ (the set of all automorphisms of $K[x]$ such that $f$ is their common eigenvector). For each polynomial $f$, the eigengroup $G_f$ is explicitly described. It is proven that every subgr…

Critical and injective modules over skew polynomial rings

Let $R$ be a commutative local $k$-algebra of Krull dimension one, where $k$ is a field. Let $α$ be a $k$-algebra automorphism of $R$, and define $S$ to be the skew polynomial algebra $R[θ; α]$. We offer, under some additional assumptions on $R$, a criterion for $S$ to have injective hulls of all simple $S$-modules locally Artinian - that is, for $S$ to satisfy property $(\diamond)$. It is easy and well known that if $α$ is of finite order, then $S$ has this property, but in order to get the criterion when $α$ has infinite order we found it necessary to classify all cyclic (Krull) critical $S$…

Some homological properties of skew PBW extensions

We prove that if R is a left Noetherian and left regular ring then the same is true for any bijective skew PBW extension A of R. From this we get Serre's Theorem for such extensions. We show that skew PBW extensions and its localizations include a wide variety of rings and algebras of interest for modern mathematical physics such as PBW extensions, well known classes of Ore algebras, operator algebras, diffusion algebras, quantum algebras, quadratic algebras in 3-variables, skew quantum polynomials, among many others. We estimate the global, Krull and Goldie dimensions, and also Quillen's K-gr…

带算子的阿贝尔群的 Ore 扩张

Ore Extensions of Abelian Groups with Operators

本文为带算子的阿贝尔群引入 Ore 群扩张，推广 Vandermonde 与 Leibniz 恒等式并建立结合性判据，在弱 $s$-幺元条件下证明 Hilbert 基定理，并应用于左模及非结合环情形。

Gabriel 商环

Gabriel Quotient Rings

本文对素 Noetherian 环 R 构造了 m-Gabriel 商环 R(m)，证明它是 Goldie 商环的子环，并刻画了其单位集与 m-full 素理想及右 Ore 集的关系。

伽罗瓦环：环论性质及其在库仑分支和仿射Hecke代数中的应用

Galois Rings: Ring-Theoretic Properties and Applications to Coulomb Branches and Affine Hecke Algebras

本文研究伽罗瓦环的环论性质，并将其应用于仿射Hecke代数和球面库仑分支代数，证明其Jacobson半本原性、满足零点定理，并计算相关维数及PI性质。