Related papers: Complete intersections of dimension zero: variatio…

Let T be a complete local (Noetherian) ring and let A be a local subring of T such that the completion of A with respect to its maximal ideal is T. We investigate the possible structures of the partially ordered set Spec(A). Specifically,…

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d \geq 2$. An $\mathfrak{m}$-primary ideal $I$ is said to be a generalized Narita ideal if $e_i^I(A) = 0$ for $2 \leq i \leq d$. If $I$ is a generalized Narita ideal and…

In this paper, we introduce a new notion, called the integral dimension, for noetherian rings. It can be regarded as the weak Briancon-Skoda numbers of rings. The point is that every noetherian local ring has finite integral dimension.

We study the linkage classes of homogeneous ideals in polynomial rings. An ideal is said to be homogeneously licci if it can be linked to a complete intersection using only homogeneous regular sequences at each step. We ask a natural…

A minor is principal means it is defined by the same row and column indices. Let $X$ be a square generic matrix, $K[X]$ the polynomial ring in entries of $X$, over an algebraically closed field, $K$. For fixed $t\leq n$, let $\mathfrak P_t$…

A proper ideal $I$ in a commutative ring with unity is called a $z^\circ$-ideal if for each $a$ in $I$, the intersection of all minimal prime ideals in $R$ which contain $a$ is contained in $I$. For any totally ordered field $F$ and a…

Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, by using the classes $ \mathcal{P}_C $ and $ \mathcal{I}_C $, we extend the notions of perfect and coperfect modules introduced by D.Rees \cite{R} and…

Let C be a commutative noetherian domain, G be a finitely generated abelian group which acts on C and B = C#G be the skew group ring. For a prime ideal I in C, we study the largest subring of B in which the right ideal IB becomes a…

In this paper, we investigate three problems concerning the toric ideal associated to a matroid. Firstly, we list all matroids $\mathcal M$ such that its corresponding toric ideal $I_{\mathcal M}$ is a complete intersection. Secondly, we…

Let $R$ be a commutative ring with nonzero identity, and $\delta :\mathcal{I(R)}\rightarrow\mathcal{I(R)}$ be an ideal expansion where $\mathcal{I(R)}$ the set of all ideals of $R$. In this paper, we introduce the concept of…

Among reduced Noetherian prime characteristic commutative rings, we prove that a regular ring is precisely one where finite intersection of ideals commutes with taking bracket powers. However, reducedness is essential for this equivalence.…

Let $T$ be a complete local (Noetherian) ring. For each $i \in \mathbb{N}$, let $C_i$ be a nonempty countable set of nonmaximal pairwise incomparable prime ideals of $T$, and suppose that if $i \neq j$, then either $C_i = C_j$ or no element…

We consider homogeneous binomial ideals $I=(f_1,\ldots,f_n)$ in $K[x_1, \ldots, x_n]$, where $f_i = a_i x_i^{d_i} - b_i m_i$ and $a_i \neq 0$. When such an ideal is a complete intersection, we show that the monomials which are not divisible…

It is shown that if $A$ is a regular local ring and $I$ is a maximally differential ideal in $A$, then $I$ is generated by an $A$-sequence.

In [1], finite associative rings wih identity and such that the set of all zero-divisors form and ideal M, called the Jacobson Radical, of cube zero and square non-zero, were constructed for all the characteristics. These rings are…

Let $(R, \mathfrak m)$ be a commutative noetherian local ring and $I$ an ideal of $R$. For every $R$-module $M$, $\gamma_I(M) = \sum\{ \operatorname{Bi} f \,|\, f \in \operatorname{Hom}_R(I,M)\}$ is called the trace of $I$ in $M$. It is…

We provide a structure theorem for all almost complete intersection ideals of depth three in any Noetherian local ring. In particular, we find that the minimal generators are the pfaffians of suitable submatrices of an alternating matrix.…

Let $S$ be a polynomial ring over an algebraic closed field $k$ and $ \mathfrak p =(x,y,z,w) $ a homogeneous height four prime ideal. We give a finite characterization of the degree two component of ideals primary to $\mathfrak p$, with…

In this paper we will investigate commutative rings which have the $\ast $-property. We say that a ring $R$ satisfy $\ast-$property if for any family of ideals $\left\{ I_{\alpha}\right\} _{\alpha\in S}$ of $R$ in which $S$ is an index set,…

We examine the ideal $I=(x_1^2, \dots, x_n^2, (x_1+\dots+x_n)^2)$ in the polynomial ring $Q=k[x_1, \dots, x_n]$, where $k$ is a field of characteristic zero or greater than $n$. We also study the Gorenstein ideal $G$ linked to $I$ via the…