A note on a new ideal

关于Ferrers图的边理想的一个注记

A note on the edge ideals of Ferrers graphs

我们确定了Ferrers图的每一个边理想的算术秩。

Notes on a minimal set of generators for the radical ideal defining the diagonal locus of $(\C^2)^n$

We develop several techniques for the study of the radical ideal $I$ defining the diagonal locus of $(\C^2)^n$. Using these techniques, we give combinatorial construction of generators for $I$ of certain bi-degrees.

A Note on Co-Maximal Ideal Graph of Commutative Rings

Let $R$ be a commutative ring with unity. The co-maximal ideal graph of $R$, denoted by $Γ(R)$, is a graph whose vertices are the proper ideals of $R$ which are not contained in the Jacobson radical of $R$, and two vertices $I_1$ and $I_2$ are adjacent if and only if $I_1 + I_2 = R$. We classify all commutative rings whose co-maximal ideal graphs are planar. In 2012 the following question was posed: If $Γ(R)$ is an infinite star graph, can $R$ be isomorphic to the direct product of a field and a local ring? In this paper, we give an affirmative answer to this question.

A note on stability properties of powers of polymatroidal ideals

Let $I$ be a matroidal ideal of degrre $d$ of a polynomial ring $R=K[x_1,...,x_n]$, where $K$ is a field. Let astab$(I)$ and dstab$(I)$ be the smallest integer $n$ for which Ass$(I^n)$ and depth$(I^n)$ stabilize, respectively. In this paper, we show that astab$(I)=1$ if and only if dstab$(I)=1$. Moreover, we prove that if $d=3$, then ${\rm astab}(I)={\rm dstab}(I)$. Furthermore, we show that if $I$ is an almost square-free Veronese type ideal of degree $d$, then ${\rm astab}(I)={\rm dstab}(I)=\lceil\frac{n-1}{n-d}\rceil$.

关于图的 toric 理想与 Knutson-Miller-Yong 分解的注记

A note on toric ideals of graphs and Knutson-Miller-Yong decompositions

该研究使用 Knutson-Miller-Yong 提出的 Gröbner 基技术，分析图性质与对应 toric 理想的代数性质的关联，恢复了图 toric 理想的高度公式，提出检测二分图删除的代数性质，并利用初始理想界定色数。

A Note on Congruences of Infinite Bounded Involution Lattices

We prove that an infinite (bounded) involution lattice and even pseudo--Kleene algebra can have any number of congruences between $2$ and its number of elements or equalling its number of subsets, regardless of whether it has as many ideals as elements or as many ideals as subsets; consequently, the same holds for antiortholattices. Under the Generalized Continuum Hypothesis, this means that an infinite (bounded) involution lattice, pseudo--Kleene algebra or antiortholattice can have any number of congruences between $2$ and its number of subsets, regardless of its number of ideals.