An ideal I in a commutative Noetherian ring R has the strong persistence property if (It+1:RI)=It for all positive integers t. In this article, we focus on how some graph operations affect the stro...

Associated primes of powers of cover ideals under graph operations

We produce a procedure for constructing new normal monomial ideals from other ideals that are assumed to be normal. This enables us to prove that if the cover ideal of a graph G is normal, then the cover ideal of the graph H is normal as well, where the graph H is obtained by connecting all vertices in G with a new vertex. We use these ideas to explore the normality of the cover ideals of some imperfect graphs. Also, we investigate the normality under expansion, this leads us to generalize the work of Al-Ayyoub (Rocky Mountain J Math 39:1–9, 2009). Furthermore, we investigate the normality under more operations such as weighting, polarization, localization, contraction, and deletion.

Normality of Cover Ideals of Graphs and Normality Under Some Operations

In this paper, we find some basic results of Multiplication Lattice Modules.

/pdf/on-multiplication-lattice-modules-23jx1z564t.pdf

On Multiplication Lattice Modules

We study the closed neighborhood ideals and the dominating ideals of graphs, in particular, of trees and cycles. We prove that the closed neighborhood ideals and the dominating ideals of trees are normally torsion-free. The closed neighborhood ideals and the dominating ideals of cycles fail to be normally torsion-free. However, we prove that the closed neighborhood ideals of cycles admit the (strong) persistence property and the dominating ideals of cycles are nearly normally torsion-free.

Dominating ideals and closed neighborhood ideals of graphs

Let $I$ be a square-free monomial ideal in a polynomial ring $R=K[x_1,\ldots, x_n]$ over a field $K$, $\mathfrak{m}=(x_1, \ldots, x_n)$ be the graded maximal ideal of $R$, and $\{u_1, \ldots, u_{\beta_1(I)}\}$ be a maximal independent set of minimal generators of $I$ such that $\mathfrak{m}\setminus x_i \notin \mathrm{Ass}(R/(I\setminus x_i)^t)$ for all $x_i\mid \prod_{i=1}^{\beta_1(I)}u_i$ and some positive integer $t$, where $I\setminus x_i$ denotes the deletion of $I$ at $x_i$ and $\beta_1(I)$ denotes the maximum cardinality of an independent set in $I$. In this paper, we prove that if $\mathfrak{m}\in \mathrm{Ass}(R/I^t)$, then $t\geq \beta_1(I)+1$. As an application, we verify that under certain conditions, every unmixed Konig ideal is normally torsion-free, and so has the strong persistence property. In addition, we show that every square-free transversal polymatroidal ideal is normally torsion-free. Next, we state some results on the corner-elements of monomial ideals. In particular, we prove that if $I$ is a monomial ideal in a polynomial ring $R=K[x_1, \ldots, x_n]$ over a field $K$ and $z$ is an $I^t$-corner-element for some positive integer $t$ such that $\mathfrak{m}\setminus x_i \notin \mathrm{Ass}(I\setminus x_i)^t$ for some $1\leq i \leq n$, then $x_i$ divides $z$.

On the embedded associated primes of monomial ideals

Let I and J be two ideals in a commutative Noetherian ring S. We say that J is a superficial ideal for I if the following conditions are satisfied: (i)G(J) ⊆ G(I), where G(L) denotes a minimal set of generators of an ideal L. (ii)(Ik+1 : SJ) = Ik for all positive integers k. In this paper, by using some monomial operators, we first introduce several methods for constructing new ideals which have superficial ideals. In the sequel, we present some examples of monomial ideals which have superficial ideals. Next, we discuss on the relation between superficiality and normality. Finally, we explore the relation between normally torsion-freeness and superficiality.

Superficial ideals for monomial ideals

Let I be a monomial ideal in a polynomial ring $$R=K[x_1, \ldots , x_n]$$
 and $${\mathfrak {m}}=(x_1, \ldots , x_n)$$
 be the graded maximal ideal of R In this paper, we provide a necessary and sufficient condition whether $${\mathfrak {m}}\in \mathrm {Ass}_R(R/I)$$
 , where $$\mathrm {Ass}_R(R/I)$$
 denotes the set of associated prime ideals of I

Detecting the maximal-associated prime ideal of monomial ideals

Let M be a faithful multiplication and comultiplication module over a commutative ring R. In this paper we investigate some results on such modules. Mathematics Subject Classification: 13C05, 13C13, 13A15 , 13C99

/pdf/some-results-on-comultiplication-modules-4tl2cbhkx7.pdf

Some Results on Comultiplication Modules

In this paper we investigate some results of multiplication modules on arithmetical rings. We introduce the concept of quasi-copure submodule of an R-module M and we will give some properties of them.

/pdf/multiplication-modules-on-arithmetical-rings-2k72a9f1ca.pdf

Multiplication Modules on Arithmetical Rings

The purpose of this paper is to explore some basic facts from radical of submodules in the free multiplication R-module M = R n .

/pdf/some-remarks-on-free-multiplication-modules-4cg6c6jrpf.pdf

Some Remarks on Free Multiplication Modules

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