C4*模 over交换环的局部化、局部-全局转移和壳理论

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 $S$…

Simple modules and their essential extensions for skew polynomial rings

Let $R$ be a commutative Noetherian ring and $α$ an automorphism of $R$. This paper addresses the question: when does the skew polynomial ring $S = R[θ; α]$ satisfy the property $(\diamond)$, that for every simple $S$-module $V$ the injective hull $E_S(V)$ of $V$ has all its finitely generated submodules Artinian. The question is largely reduced to the special case where $S$ is primitive, for which necessary and sufficient conditions are found, which however do not between them cover all possibilities. Nevertheless a complete characterisation is found when $R$ is an affine algebra over a field…

A Note on Coseparable Coalgebras

Given a coalgebra $C$ over a commutative ring $R,$ we show that $C$ can be considered as a (not necessarily counital) $C^{\ast op}$-coring. Moreover, we show that this coring has a left (right) counity if and only if $C$ is coseparable as an $R$-coalgebra.

FP-injectivity of factors of injective modules

It is shown that a ring is left semihereditary if and only each homomorphic image of its injective hull as left module is FP-injective. It is also proven that a commutative ring R is reduced and arithmetical if and only if E/U if FP-injective for any FP-injective R-module E and for any submodule U of finite Goldie dimension. A characterization of commutative rings for which each module of finite Goldie dimension is of injective dimension at most one is given. Let R be a chain ring and Z its subset of zerodivisors. It is proven that E/U is FP-injective for each FP-injective R-module E and each …

Maximal quadratic modules on *-rings

We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to $\ast$-rings and discuss the relation of this generalization to recent developments in noncommutative real algebraic geometry. The simplest example of a maximal proper quadratic module is the cone of all positive semidefinite complex matrices of a fixed dimension. We show that the support of a maximal proper quadratic module is the symmetric part of a prime $\ast$-ideal, that every maximal proper quadratic module in a Noetherian $\ast$-ring comes from a maximal proper quadratic module i…

关于交换环零因子图特征值的一个注记

A note on eigenvalues of zero divisor graphs associated with commutative rings

本文纠正了交换环零因子图特征值、能量及Zagreb指标已有结果的错误，并证明特定环的零因子图满足相应能量性质和Hansen-Vukičević猜想。

关于线性递推序列的若干代数结果

Some algebraic results concerning linear recurrence sequences

本文研究由分裂首一多项式对应的线性递推序列空间构成的集合，赋予其两种分次交换半环结构，并紧凑刻画序列空间乘积与星积所对应的多项式。

C4*模的精确序列稳定性与环境实现

Smith-Orbit Classification of Extensions and Exact-Sequence Asymmetry for C4*-Modules

本文研究C4*模的精确序列稳定性，提出稳定条件并探讨其在子模和商模中的应用，同时构建了适合求商敏感模理论的环境框架。

二次单项式代数 ${\rm N}_m$ 的 Hochschild 上同调

Hochschild cohomology of the quadratic monomial algebra ${\rm N}_m$

本文计算了交换环上二次单项式代数 ${\rm N}_m(R)$ 的 Hochschild 上同调模与上同调环结构，并证明当 $m \ge 3$ 时该上同调环为无限生成且不具有给出 Gerstenhaber 括号的 Batalin-Vilkovisky 代数结构。

当R不满足pp-可定义子群的DCC时，R-模的Borel完备性

Borel completeness of $R$-modules when $R$ fails the DCC on pp-definable subgroups

该研究证明了非左完全可数环的R-模理论是Borel完备的，刻画了可数单环的Borel完备性，引入了相关理想与包络构造，强化了TFAB的Borel完备性结论。

提升 Leavitt 路代数分次 Grothendieck 群之间的态射

Lifting morphisms between graded Grothendieck groups of Leavitt path algebras

本文证明有限图 Bowen-Franks 模间的预序模映射可提升为 Leavitt 路代数间的保对角分次 $*$-同态，从而在域情形证实 Hazrat 猜想的满性部分。