On the dot product graph of a commutative ring, II

复曲面拓扑。第三章面环的交换与同调代数

Toric Topology. Chapter 3: Commutative and homological algebra of face rings

这是我们的“Toric Topology”图书项目的第三章。更多的章节即将到来。欢迎提出意见和建议。

A graph which recognizes idempotents of a commutative ring

In this paper we introduce and study a graph on the set of ideals of a commutative ring $R$. The vertices of this graph are non-trivial ideals of $R$ and two distinct ideals $I$ and $J$ are adjacent if and only $IJ=I\cap J$. We obtain some properties of this graph and study its relation to the structure of $R$.

Computing the strong metric dimension for co-maximal ideal graphs of commutative rings

Let $R$ be a commutative ring with identity. The co-maximal ideal graph of $R$, denoted by $Γ(R)$, is a simple graph whose vertices are proper ideals of $R$ which are not contained in the Jacobson radical of $R$ and two distinct vertices $I, J$ are adjacent if and only if $I+J=R$. In this paper, we use Gallai$^{^,}$s Theorem and the concept of strong resolving graph to compute the strong metric dimension for co-maximal ideal graphs of commutative rings. Explicit formulae for the strong metric dimension, depending on whether the ring is reduced or not, are established.

有限交换环上的向量空间

Vector spaces over finite commutative rings

本文推广有限域上的向量空间研究至有限交换环，获得子空间的计数公式和维数公式，并应用于讨论弧和帽。

有限交换环上的量子分数复兴

Quantum fractional revival on unitary Cayley graphs over finite commutative rings

本文研究了有限交换环上的单位Cayley图中量子分数复兴的存在性，并刻画了允许此类复兴的环结构。

Existence of Erdős-Burgess constant in commutative rings

Let $R$ be a commutative unitary ring. An idempotent in $R$ is an element $e\in R$ with $e^2=e$. The Erdős-Burgess constant associated with the ring $R$ is the smallest positive integer $\ell$ (if exists) such that for any given $\ell$ elements (not necessarily distinct) of $R$, say $a_1,\ldots,a_{\ell}\in R$, there must exist a nonempty subset $J\subset \{1,2,\ldots,\ell\}$ with $\prod\limits_{j\in J} a_j$ being an idempotent. In this paper, we prove that except for an infinite commutative ring with a very special form, the Erdős-Burgess constant of the ring $R$ exists if and only if $R$ is f…

具有几乎交换Terwilliger代数的循环群上的Schur环

Schur rings over cyclic groups having Almost Commutative Terwilliger algebras

分类有限循环群上产生几乎交换Terwilliger代数的轨道Schur环，并给出Schur环的楔积产生几乎交换Terwilliger代数的条件。

Covering numbers of commutative rings

A cover of a unital, associative (not necessarily commutative) ring $R$ is a collection of proper subrings of $R$ whose set-theoretic union equals $R$. If such a cover exists, then the covering number $σ(R)$ of $R$ is the cardinality of a minimal cover, and a ring $R$ is called $σ$-elementary if $σ(R) < σ(R/I)$ for every nonzero two-sided ideal $I$ of $R$. In this paper, we show that if $R$ has a finite covering number, then the calculation of $σ(R)$ can be reduced to the case where $R$ is a finite ring of characteristic $p$ and the Jacobson radical $J$ of $R$ has nilpotency 2. Our main result…

交换环的素理想和图的强度量维数

Strong metric dimension of the prime ideal sum graph of a commutative ring

本文研究含单位元交换环的素理想和图，并确定了多类 Artin 非局部交换环对应图的强度量维数。