Noncrossed Product Matrix Subrings and Ideals of Graded Rings

Shephard-Todd-Chevalley Theorem for skew polynomial rings

We prove the following generalization of the classical Shephard-Todd-Chevalley Theorem. Let $G$ be a finite group of graded algebra automorphisms of a skew polynomial ring $A:=k_{p_{ij}}[x_1,...,x_n]$. Then the fixed subring $A^G$ has finite global dimension if and only if $G$ is generated by quasi-reflections. In this case the fixed subring $A^G$ is isomorphic a skew polynomial ring with possibly different $p_{ij}$'s. A version of the theorem is proved also for abelian groups acting on general quantum polynomial rings.

Reconstruction of graded groupoids from graded Steinberg algebras

We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies $C^*$-isomorphism of $C^*$-algebras for graphs $E$ and $F$ in which every cycle has an exit.

Simple Semigroup Graded Rings

We show that if $R$ is a, not necessarily unital, ring graded by a semigroup $G$ equipped with an idempotent $e$ such that $G$ is cancellative at $e$, the non-zero elements of $eGe$ form a hypercentral group and $R_e$ has a non-zero idempotent $f$, then $R$ is simple if and only if it is graded simple and the center of the corner subring $f R_{eGe} f$ is a field. This is a generalization of a result of E. Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity o…

Bimodules in group graded rings

In this article we introduce the notion of a controlled group graded ring. Let $G$ be a group, with identity element $e$, and let $R=\oplus_{g\in G} R_g$ be a unital $G$-graded ring. We say that $R$ is $G$-controlled if there is a one-to-one correspondence between subsets of the group $G$ and (mutually non-isomorphic) $R_e$-bimodules in $R$, given by $G \supseteq H \mapsto \oplus_{h\in H} R_h$. For strongly $G$-graded rings, the property of being $G$-controlled is stronger than that of being simple. We provide necessary and sufficient conditions for a general $G$-graded ring to be $G$-controll…

由有序群分次的非单位素环

Nonunital prime rings graded by ordered groups

该研究将有序群分次素环的经典定理推广至非单位环情形，建立分次素理想与单位分支素理想的双射对应，并给出Leavitt路径环素性新证明及群环分次子环素性条件。

$\mathbb{Z}$-graded rings as Cuntz-Pimsner rings

Given a $\mathbb{Z}$-graded ring $A$ and a subring $R\subseteq A$, it is natural to ask whether $A$ can be realised as the Cuntz-Pimsner ring of some $R$-system. In this paper, we derive sufficient conditions on $A$ and $R$ for this to be the case. As an application, we give conditions under which the Steinberg algebra $A_K(\mathcal{G})$ associated to a $\mathbb{Z}$-graded groupoid $\mathcal{G}=\sqcup_{n\in \mathbb{Z}} \mathcal{G}_n$ can be realised as the Cuntz-Pimsner ring of an $A_K(\mathcal{G}_0)$-system.

Simple group graded rings and maximal commutativity

In this paper we provide necessary and sufficient conditions for strongly group graded rings to be simple. For a strongly group graded ring $R = \bigoplus_{g\in G} R_g$ the grading group $G$ acts, in a natural way, as automorphisms of the commutant of the neutral component subring $R_e$ in $R$ and of the center of $R_e$. We show that if $R$ is a strongly $G$-graded ring where $R_e$ is maximal commutative in $R$, then $R$ is a simple ring if and only if $R_e$ is $G$-simple (i.e. there are no nontrivial $G$-invariant ideals). We also show that if $R_e$ is commutative (not necessarily maximal com…

Epsilon-strongly graded rings: Azumaya algebras and partial crossed products

The main purpose of this paper is to investigate epsilon-strongly graded rings that are partial crossed products. Let $G$ be a group, $A=\oplus_{g\in G}\,A_g$ an epsilon-strongly graded ring and ${\bf pic}{R}$ the Picard semigroup of $R:=A_1$. We prove that the isomorphism class $[A_g]$ is an element of ${\bf pic}{R}$, for all $g\in G$. Thus, the association $g\mapsto [A_g]$ determines a partial representation of $G$ on ${\bf pic}{R}$ which induces a partial action $γ$ of $G$ on the center $Z(R)$ of $R$. Sufficient conditions for $A$ to be an Azumaya $R^γ$-algebra are presented in the case tha…

分次微分多项式环

Graded differential polynomial rings

研究分次环上的微分多项式环何时具有相容分次，并建立其分次版本的简单性、素性和Noether性定理。

关于Burnside环的全幂运算的像

On the image of the total power operation for Burnside rings

该研究证明Burnside环全幂运算的像落在组合子环内，构造了交换分次环并给出特征公式，推广了Frobenius–Wielandt同态和Burnside轨道计数引理。

Ideal growth in amalgamated powers of nilpotent rings of class two and zeta functions of quiver representations

Let $L$ be a nilpotent algebra of class two over a compact discrete valuation ring $A$ of characteristic zero or of sufficiently large positive characteristic. Let $q$ be the residue cardinality of $A$. The ideal zeta function of $L$ is a Dirichlet series enumerating finite-index ideals of $L$. We prove that there is a rational function in $q$, $q^m$, $q^{-s}$, and $q^{-ms}$ giving the ideal zeta function of the amalgamation of $m$ copies of $L$ over the derived subring, for every $m \geq 1$, up to an explicit factor. More generally, we prove this for the zeta functions of nilpotent quiver rep…

Maximal Crossed Product Orders over Discrete Valuation Rings

The problem of determining when a (classical) crossed product $T=S^f*G$ of a finite group $G$ over a discrete valuation ring $S$ is a maximal order, was answered in the 1960's for the case where $S$ is tamely ramified over the subring of invariants $S^G$. The answer was given in terms of the conductor subgroup (with respect to $f$) of the inertia. In this paper we solve this problem in general when $S/S^G$ is residually separable. We show that the maximal order property entails a restrictive structure on the sub-crossed product graded by the inertia subgroup. In particular, the inertia is abel…

Normality of algebras over commutative rings, crossed pairs, and the Teichmueller class

Let S be a commutative ring, Q a group that acts on S, and let R be the subring of S fixed under Q. A Q-normal S-algebra consists of a central S-algebra A and a homomorphism s from Q to the group Out(A) of outer automorphisms of A that lifts the Q-action on S. We associate to a Q-normal S-algebra (A,s) a crossed 2-fold extension which, in turn, represents a class, the Teichmueller class of (A,s), in the third cohomology group of Q with coefficients in the group U(S) of units of S, endowed with the obvious Q-module structure. Suitable equivalence classes of Q-normal Azumaya S-algebras constitut…