K. Ardakov, K. A. Brown
This is a survey of the known properties of Iwasawa algebras, which are completed group rings of compact p-adic analytic groups with coefficients the ring Zp of p-adic integers or the field Fp of p elements. A number of open questions are also stated.
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Total rings of quotients of higher dimensional Iwasawa algebras
该文将 Ritter--Weiss 等人关于 $d=1$ 时 Iwasawa 代数全商环的 Wedderburn 分解结果推广到任意维数 $d\ge1$，研究对象为有限群与 $\mathbb Z_p^d$ 的半直积对应的 Iwasawa 代数。
On the Wedderburn decomposition of the total ring of quotients of certain Iwasawa algebras
本文研究了某些Iwasawa代数的完整商环的Wedderburn分解，通过与群环的分解联系，在分枝假设下确定其结构。
Let $p>2$ be a prime, $k$ a finite field of characteristic $p$, and $G$ a nilpotent-by-finite compact $p$-adic analytic group. Write $kG$ for the completed group ring of $G$ over $k$. We show that $kG$ is a catenary ring.
On the Wedderburn decomposition of the total ring of quotients of certain Iwasawa algebras II
本文研究了某些Iwasawa代数的总商环的Wedderburn分解，通过群环的分解来全面确定其结构。
Let $k$ be a finite field of characteristic $p$, and $G$ a compact $p$-adic analytic group. Write $kG$ for the completed group ring of $G$ over $k$. In this paper, we describe the structure of the ring $kG/P$, where $P$ is a minimal prime ideal of $kG$. We give an isomorphism between $kG/P$ and a matrix ring with coefficients in the ring $(k'G')_α$, where $k'/k$ is a finite field extension, $G'$ is a large subquotient of $G$ with no finite normal subgroups, and $(-)_α$ is a "twisting" operation that preserves several desirable properties of the ring structure. We demonstrate an application of …
We study prime ideals in skew power series rings $T:=R[[y;τ,δ]]$, for suitably conditioned right noetherian complete semilocal rings $R$, automorphisms $τ$ of $R$, and $τ$-derivations $δ$ of $R$. These rings were introduced by Venjakob, motivated by issues in noncommutative Iwasawa theory. Our main results concern "Cutting Down" and "Lying Over." In particular, under the additional assumption that $δ= τ- id$ (a basic feature of the Iwasawa-theoretic context), we prove: If $I$ is an ideal of $R$, then there exists a prime ideal $P$ of $S$ contracting to $I$ if and only if $I$ is a $δ$-stable $τ…
This paper is a natural continuation of the study of skew power series rings A initiated in [P. Schneider and O. Venjakob, On the codimension of modules over skew power series rings with applications to Iwasawa algebras, J. Pure Appl. Algebra 204 (2005), 349 - 367.]. We construct skew Laurent series rings B and show the existence of some canonical Ore sets S for the skew power series rings A such that a certain completion of the localisation A_S is isomorphic to B. This is applied to certain Iwasawa algebras. Finally we introduce subrings of overconvergent skew Laurent series rings.
Let G be a nilpotent complete p-valued group of finite rank and let k be a field of characteristic p. We prove that every faithful prime ideal of the Iwasawa algebra kG is controlled by the centre of G, and use this to show that the prime spectrum of kG is a disjoint union of commutative strata. We also show that every prime ideal of kG is completely prime. The key ingredient in the proof is the construction of a non-commutative valuation on certain filtered simple Artinian rings.
Let $G$ be a nilpotent-by-finite compact $p$-adic analytic group for some $p>2$, and $H = \mathbf{FN}_p(G)$ its finite-by-(nilpotent $p$-valuable) radical. Fix a finite field $k$ of characteristic $p$, and write $kG$ for the completed group ring of $G$ over $k$. We show that almost faithful $G$-stable prime ideals $P$ of $kH$ extend to prime ideals $PkG$ of $kG$.
Necessary and sufficient conditions are given for the completed group algebras of a compact p-adic analytic group with coefficient ring the p-adic integers or the field of p elements to be prime, semiprime and a domain. Necessary and sufficient conditions for the localisation at semiprime ideals related to the augmentation ideals of closed normal subgroups are found. Some information is obtained about the Krull and global dimensions of the localisations. The results extend and complete work of A. Neumann and J. Coates et al.
Skew power series rings with automorphisms of finite inner order
本文研究正特征标准过滤阿廷代数上的有界斜幂级数环，将 σ 无限内阶时的单性结果推广到有限内阶情形，证明其通常单且总是素，并解决 Letzter 问题的重要特例。
Let $B=A[[t;σ,δ]]$ be a skew power series ring such that $σ$ is given by an inner automorphism of $B$. We show that a certain Waldhausen localisation sequence involving the K-theory of $B$ splits into short split exact sequences. In the case that $A$ is noetherian we show that this sequence is given by the localisation sequence for a left denominator set $S$ in $B$. If $B=Z_p[[G]]$ happens to be the Iwasawa algebra of a $p$-adic Lie group $G\isomorph H\rtimes Z_p$, this set $S$ is Venjakob's canonical Ore set. In particular, our result implies that $$ 0--> K_{n+1}(Z_p[[G]])--> K_{n+1}(Z_p[[G]]…
On group rings of the simple group of order 168, 504 or 360 and their modules
本文研究阶为168、504或360的单群G的群环模的χ-部分结构，这些模源自理想类群、Artin L-函数和Iwasawa理论。
The first purpose of this paper is to set up a general notion of skew power series rings S over a coefficient ring R, which are then studied by filtered ring techniques. The second subject consists of investigating the class of S-modules which are finitely generated as R-module. In the case that S and R are Auslander regular we show in particular that the codimension of M as S-module is one higher than the codimension of M as R-module. Furthermore its class in the Grothendieck group of S-modules of codimension at most one less vanishes, which is in the spirit of the Gersten conjecture for comm…
In this paper, we investigate the structure of skew power series rings of the form $S = R[[x;σ,δ]]$, where $R$ is a complete filtered ring and $(σ,δ)$ is a skew derivation respecting the filtration. Our main focus is on the case in which $σδ= δσ$, and we aim to use techniques in non-commutative valuation theory to address the long-standing open question: if $P$ is an invariant prime ideal of $R$, is $PS$ a prime ideal of $S$? When $R$ has characteristic $p$, our results reduce this to a finite-index problem. We also give preliminary results in the "Iwasawa algebra" case $δ= σ- \mathrm{id}_R$ i…
The canonical dimension of modules for Iwasawa algebras
本文研究Iwasawa代数中模块的规范维度，证明了其无规范维度为一的模块，并给出了Iwasawa代数的Krull维度新上界，同时提出了一项关于有限长度表示的规范维度准则。
We investigate the vanishing of the group SK1(A(G)) for the Iwasawa algebra A(G) of a pro-p p-adic Lie group G (with p different from 2). We reduce this vanishing to a linear algebra problem for Lie algebras over arbitrary rings, which we solve for Chevalley orders in split reductive Lie algebras.
Let $α=(A_g,α_g)_{g\in G}$ be a group-type partial action of a connected groupoid $G$ on a ring $A=\bigoplus_{z\in G_0}A_z$ and $B=A\star_αG$ the corresponding partial skew groupoid ring. In the first part of this paper we investigate the relation of several ring theoretic properties between $A$ and $B$. For the second part, using that every Leavitt path algebra is isomorphic to a partial skew groupoid ring obtained from a partial groupoid action $λ$, we characterize when $λ$ is group-type. In such a case, we obtain ring theoretic properties of Leavitt path algebras from the results on general…
In this paper we work with unital twisted partial actions. We investigate ring theoretic properties of partial crossed products as artinianity, noetherianity, perfect property, semilocalproperty, semiprimary property and we also study the Krull dimension. Moreover, we consider triangular matrix representation of partial skew group rings, weak and global dimensions of partial crossed products Also we study when the partial crossed products are Frobenius and symmetric algebras.
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。
We associate to every quandle $X$ and an associative ring with unity $\mathbf{k}$, a nonassociative ring $\mathbf{k}[X]$ following [3]. The basic properties of such rings are investigated. In particular, under the assumption that the inner automorphism group $Inn(X)$ acts orbit $2$-transitively on $X$, a complete description of right (or left) ideals is provided. The complete description of right ideals for the dihedral quandles $R_n$ is given. It is also shown that if for two quandles $X$ and $Y$ the inner automorphism groups act $2$-transitively and $\mathbf{k}[X]$ is isomorphic to $\mathbf{…
Let $G$ be a compact $p$-adic analytic group. We recall the well-understood finite radical $Δ^+$ and FC-centre $Δ$, and introduce a $p$-adic analogue of Roseblade's subgroup $\mathrm{nio}(G)$, the unique largest orbitally sound open normal subgroup of $G$. Further, when $G$ is nilpotent-by-finite, we introduce the finite-by-(nilpotent $p$-valuable) radical $\mathbf{FN}_p(G)$, an open characteristic subgroup of $G$ contained in $\mathrm{nio}(G)$. By relating the already well-known theory of isolators with Lazard's notion of $p$-saturations, we introduce the isolated lower central (resp. isolate…
A Model Theoretic Perspective on Matrix Rings
本文从模型论角度研究矩阵环，给出量词消去的充要条件，并揭示其与同步共轭问题及不可判定性的关联。
The article gives a ring theoretic perspective on cluster algebras. Geiß-Leclerc-Schröer prove that all cluster variables in a cluster algebra are irreducible elements. Furthermore, they provide two necessary conditions for a cluster algebra to be a unique factorization domain, namely the irreducibility and the coprimality of the initial exchange polynomials. We present a sufficient condition for a cluster algebra to be a unique factorization domain in terms of primary decompositions of certain ideals generated by initial cluster variables and initial exchange polynomials. As an application, t…
We introduce and study model-theoretic connected components of rings as an analogue of model-theoretic connected components of definable groups. We develop their basic theory and use them to describe both the definable and classical Bohr compactifications of rings. We then use model-theoretic connected components to explicitly calculate Bohr compactifications of some classical matrix groups, such as the discrete Heisenberg group $UT_3(Z)$, the continuous Heisenberg group $UT_3(R)$, and, more generally, groups of upper unitriangular and invertible upper triangular matrices over unital rings.
The values of the normalized homogeneous weight are determined for arbitrary finite Frobenius rings and expressed in a form that is independent from a generating character and the Möbius function on the ring. The weight naturally induces a partition of the ring, which is invariant under left or right multiplication by units. It is shown that the character-theoretic left-sided dual of this partition coincides with the right-sided dual, and even more, the left- and right-sided Krawtchouk coefficients coincide. An example is provided showing that this is not the case for general invariant partiti…
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