Keqin Liu
We introduce the notions of one-sided dirings, 3-irreducible left modules, 3-primitive left dirings, 3-semi-primitive left dirings, 3-primitive ideals and 3-radicals. The main results consists of two parts. The first part establishes two external characterizations of a 3-semi-primitive left diring. The second part characterizes the 3-radical of a left diring by using 3-primitive ideals.
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We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal) field extensions. This leads us naturally to consider non-unital groupoid graded rings of a particular type that we call object unital. We determine when such rings are strongly graded, crossed products, skew groupoid rings and twisted groupoid rings. We also obtain necessary and sufficient criteria for when object unital groupoid graded rings are separable over their principal component, thereby generalizi…
The object of this paper is to examine finite solvable groups whose integral group rings have only trivial central units.
We use the concept of a regular object with respect to another object in an arbitrary category, defined in \cite{dntd}, in order to obtain the transfer of regularity in the sense of Zelmanowitz between the categories $R-$mod and $S-$mod, when $S$ is an excellent extension of the ring $R$. Consequently, we obtain a result of \cite{ps}: if $S$ is an excellent extension of the ring $R$, then $S$ is von Neumann regular ring if and only if $R$ is also von Neumann regular ring. In the second part, using relative regular modules, we give a new proof of a classical result: the von Neumann regular prop…
On a category of chains of modules whose endomorphism rings have at most $2n$ maximal ideals
本文研究一类带固定子模链的右$R$-模构成的加法范畴，描述其自同态环性质并分析该范畴对象的直和行为。
The object of this paper is to study (infinite) groups whose integral group rings have only trivial central units. This property is closely related to a property, here called the RS-property (\cite{DMS05}, \cite{RS90}), involving conjugacy in the group.
The object of study in this paper is the finite groups whose integral group rings have only trivial central units. Prime-power groups and metacyclic groups with this property are characterized. Metacyclic groups are classified according to the central height of the unit groups of their integral group rings.
In order to study graded Frobenius algebras from a ring theoretical perspective, we introduce graded quasi-Frobenius rings, graded Frobenius rings and a shift-version of the latter ones, and we investigate the structure and representations of such objects. We need to revisit graded simple graded left Artinian rings, graded semisimple rings, and to provide graded versions of certain results concerning the Jacobson radical, the singular radical, and their connection to finiteness conditions and injectivity. We prove a structure result for (shift-)graded Frobenius rings.
Separability for relative extensions of object unital strongly groupoid graded rings
本文研究了在群oids格环的相对扩展下，环R/R_Δ的分离性条件，通过引入相对跟踪映射和中心元素，推广了关于矩阵环、群格环以及群oids格环的分离性定理。
In a recent paper of Alahmadi, Alkan and Lopez-Permouth, a ring R is defined to have no (simple) middle class if the injectivity domain of any (simple) R-module is the smallest or largest possible. Er, Lopez-Permouth and Sokmez use this idea of restricting the class of injectivity domains to classify rings, and give a partial characterization of rings with no middle class. In this work, we continue the study of the property of having no (simple) middle class. We give a structural description of right Artinian right nonsingular rings with no right middle class. We also give a characterization o…
Let $R$ be a ring, $\CC_R$ and $\pCCR$ be the set of regular and left regular elements of $R$ ($\CC_R\subseteq \pCCR$). Goldie's Theorem is a semisimplicity criterion for the classical left quotient ring $Q_{l,cl}(R):=\CC_R^{-1}R$. Semisimplicity criteria are given for the classical left regular left quotient ring $'Q_{l,cl}(R):=\pCCR^{-1}R$. As a corollary, two new semisimplicity criteria for $Q_{l,cl}(R)$ are obtained (in the spirit of Goldie).
In a previous article (see \cite{CNP}), we introduced and analyzed ring-theoretic properties of object unital $\mathcal{G}$-graded rings $R$, where $\mathcal{G}$ is a groupoid. In the present article, we analyze the category $\grmod$ of unitary $\G$-graded modules over such rings. Following ideas developed earlier by one of the authors in \cite{lundstrom2004}, we analyze the forgetful functor $U \colon \grmod \to \rmod$ and aim to determine properties $\mathcal{P}$ for which the following implications are valid for modules $M$ in $\grmod$: $M$ is $\mathcal{P}$ $\Rightarrow$ $U(M)$ is $\mathcal…
Let $R$ be an associative ring with identity and let $N$ be a nil ideal of $R$. It is shown that units of $R/N$ can be lifted to units in $R$. Under some mild conditions on the ring, a procedure is given to determine those lifted units in a recursive way. As an application, the units of several classes of rings are determined, including: matrix rings, chain rings, and group rings where the ring is a chain ring. Numerical examples are given illustrating the main results.
In this paper, we introduce a class of $J$-quasipolar rings. Let $R$ be a ring with identity. An element $a$ of a ring $R$ is called {\it weakly $J$-quasipolar} if there exists $p^2 = p\in comm^2(a)$ such that $a + p$ or $a-p$ are contained in $J(R)$ and the ring $R$ is called {\it weakly $J$-quasipolar} if every element of $R$ is weakly $J$-quasipolar. We give many characterizations and investigate general properties of weakly $J$-quasipolar rings. If $R$ is a weakly $J$-quasipolar ring, then we show that (1) $R/J(R)$ is weakly $J$-quasipolar, (2) $R/J(R)$ is commutative, (3) $R/J(R)$ is redu…
We introduce the notion of Azumaya object in general homotopy-theoretic settings. We give a self-contained account of Azumaya objects and Brauer groups in bicategorical contexts, generalizing the Brauer group of a commutative ring. We go on to describe triangulated bicategories and prove a characterization theorem for Azumaya objects therein. This theory applies to give a homotopical Brauer group for derived categories of rings and ring spectra. We show that the homotopical Brauer group of an Eilenberg-Mac Lane spectrum is isomorphic to the homotopical Brauer group of its underlying commutativ…
In a previous work we constructed the $Q$-shaped derived category of any ring $A$ for any suitably nice category $Q$. The $Q$-shaped derived category of $A$, which is denoted by $\mathcal{D}_{Q}(A)$, is a generalization of the ordinary derived category. In this paper we prove that the $Q$-shaped derived category of $A$ is a compactly generated triangulated category. We also define perfect objects in $\mathcal{D}_{Q}(A)$ and prove that these constitute a triangulated subcategory, $\mathcal{D}^\mathrm{perf}_{Q}(A)$, of the category $\mathcal{D}_{Q}(A)^\mathrm{c}$ of compact objects in the $Q$-sh…
Actions of pro-groups and pro-rings
本文探讨了pro群和非单位pro环的内部动作，通过群对象和环对象在Pro(Set)中的动作进行描述，并指出李代数中类似结果不成立。
The primary object of this paper is to prove the conjecture of the authors from a previous paper, explaining how to recover the weak dimension of a ring from its derived category. In the process, we develop a theory of weak dimension, which we call ghost dimension, for the generalized rings, known as ring spectra, that arise in algebraic topology.
Topological endomorphism rings of tilting complexes
本文在紧致生成三角范畴中引入正派倾斜对象，证明其倾斜心等价于自同态环上的反模范畴，推广了已有结果，并建立了正派倾斜复形与倾斜和余倾斜导出等价对之间的Morita理论。
We give an overview of recent developments in silting theory. After an introduction on torsion pairs in triangulated categories, we discuss and compare different notions of silting and explain the interplay with t-structures and co-t-structures. We then focus on silting and cosilting objects in a triangulated category with coproducts and study the case of the unbounded derived category of a ring. We close the survey with some classification results over commutative noetherian rings and silting-discrete algebras.
Albert algebras, a specific kind of Jordan algebra, are naturally distinguished objects among commutative non-associative algebras and also arise naturally in the context of simple affine group schemes of type $F_4$, $E_6$, or $E_7$. We study these objects over an arbitrary base ring $R$, with particular attention to the case of the integers. We prove in this generality results previously in the literature in the special case where $R$ is a field of characteristic different from 2 and 3.
We introduce and develop a structure theory of a new class of noncommutative rings - Galois orders, that generalize classical orders in noncommutative rings. Galois orders realized as certain subrings of invariants in skew semigroup rings. This class of rings contains many classical objects, in particular the Generalized Weyl algebras, the universal enveloping algebra of the general linear Lie algebra and certain rings of invariant differential operators on algebraic varieties.
For an arbitrary ring $R$, the largest strong left quotient ring $Q_l^s(R)$ of $R$ and the strong left localization radical $\glsR$ are introduced and their properties are studied in detail. In particular, it is proved that $Q_l^s(Q_l^s(R))\simeq Q_l^s(R)$, $\gll^s_{R/\glsR}=0$ and a criterion is given for the ring $ Q_l^s(R)$ to be a semisimple ring. There is a canonical homomorphism from the classical left quotient ring $Q_{l, cl}(R)$ to $Q_l^s(R)$ which is not an isomorphism, in general. The objects $Q_l^s(R)$ and $\gll^s_R$ are explicitly described for several large classes of rings (semip…
A local ring $R$ is called $Z$-local if $J(R) = Z(R)$ and $J(R)^2 = 0$. In this paper the structures of a class of $Z$-local rings are determined.
Tilting complexes and codimension functions over commutative noetherian rings
本文在交换诺特环的导出范畴中，利用局部上同调构造与 sp-滤过关联的 silting 对象，并刻画其成为 tilting 的条件，特别与余维数函数及 Cohen-Macaulay 环同态像相关，同时给出 cosilting 对偶结果。
For a $Z$-graded Gorenstein ring $R$, we study the stable category $CM^ZR$ of $Z$-graded maximal Cohen-Macaulay $R$-modules, which is canonically triangle equivalent to the singularity category of Buchweitz and Orlov. Its thick subcategory given as the stable category of $CM_0^ZR$ is central in representation theory since it enjoys Auslander-Reiten-Serre duality and has almost split triangles. In the case $dim R=1$, we prove that the stable category of $CM_0^ZR$ always admits a silting object, and that it admits a tilting object if and only if either $R$ is regular or the $a$-invariant of $R$ …
Hopkins-Levitzki Type Theorems for Groupoid Graded Rings
该研究针对群胚分次环，引入分次理想的左/右逐对象幂零性概念，推广Hopkins-Levitzki定理，证明三类Γ₀-Artinian环为Γ₀-Noetherian环，并用反例说明相关结论。
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