Rahul Kumar, Atul Gaur
We study the ring extensions R \subseteq T having the same set of prime ideals provided Nil(R) is a divided prime ideal. Some conditions are given under which no such T exist properly containing R. Using idealization theory, the examples are also discussed to strengthen the results.
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It is shown that any set of nonzero monomial prime ideals can be realized as the stable set of associated prime ideals of a monomial ideal. Moreover, an algorithm is given to compute the stable set of associated prime ideals of a monomial ideal.
We prove that if the initial ideal of a prime ideal is Borel-fixed and the dimension of the quotient ring is less than or equal to two, then given any non-minimal associated prime ideal of the initial ideal it contains another associated prime ideal of dimension one larger.
It is shown that a squarefree principal Borel ideal satisfies the persistence property for the associated prime ideals. For the graded maximal ideal we compute the index of stability with respect to squarefree principal Borel ideals and determine their stable set of associated prime ideals.
In this paper we consider the local cohomology of monomial ideals with respect to monomial prime ideals and show that all these local cohomology modules are tame.
It is known that the polyomino ideal of a simple polyomino is a prime ideal. A new class of nonsimple polyominoes $\Pc$ for which the polyomino ideal $I_{\Pc}$ is a prime ideal will be presented.
In this paper, we introduce a new generalization of weakly prime ideals called $I$-prime. Suppose $R$ is a commutative ring with identity and $I$ a fixed ideal of $R$. A proper ideal $P$ of $R$ is $I$-prime if for $a, b \in R$ with $ab \in P-IP$ implies either $a \in P$ or $b \in P$. We give some characterizations of $I$-prime ideals and study some of its properties. Moreover, we give conditions under which $I$-prime ideals becomes prime or weakly prime and we construct the view of $I$-prime ideal in decomposite rings.
In this paper, the new concept of quasi-prime ideal is introduced which at the same time generalizes the `prime ideal' and `primary ideal' notions. Then a natural topology on the set of quasi-prime ideals of a ring is introduced which generalizes the Zariski topology. The basic properties of the quasi-prime spectrum are studied and several interesting results are obtained. Specially, it is proved that if the Grothendieck t-functor is applied on the quasi-prime spectrum then the prime spectrum is deduced. It is also shown that there are the cases that the prime spectrum and quasi-prime spectrum…
In this paper, we introduce the concepts of strongly 2-absorbing primary ideals (resp., submodules) and strongly 2-absorbing ideals (resp., submodules) as generalizations of strongly prime ideals. Furthermore, we investigate some basic properties of these classes of ideals.
Quadratic linear strands of prime ideals
研究素数理想在标准 graded多项式环中的解的二次串的精确估计，证明高度为h的理想最多含h²个二次极小生成元，且存在由h²个二次式生成的理想。
Let $R$ be a commutative integral domain and let $\star$ be a semistar operation of finite type on $R$, and $I$ be a quasi-$\star$-ideal of $R$. We show that, if every minimal prime ideal of $I$ is the radical of a $\star$-finite ideal, then the set $\Min(I)$ of minimal prime ideals over $I$ is finite.
This paper investigates the doubly exponential ideal membership property of the Mayr-Meyer ideals from the point of view of their associated primes. A doubly-exponential upper bound on the set of associated primes is proved. In the paper a new family of ideals emerges which also has the doubly exponential ideal membership property. More on the new family can be found in author's paper "A new family of ideals with the doubly exponential ideal membership property".
In this paper we present a technical lemma about localization at countable infinitely many prime ideals. We apply this lemma to get many results about the finiteness of associated prime ideals of local cohomology modules.
We prove that the initial ideal of the defining ideal of a monomial curve that corresponds to an almost arithmetic sequence of positive integers is Ratliff-Rush closed.
This paper introduce and study weakly 1-absorbing prime ideals in commutative rings. Let $A$ be a commutative ring with a nonzero identity $1\neq 0$. A proper ideal $P$ of $A$ is said to be a weakly 1-absorbing prime ideal if for each nonunits $x, y, z \in A$ with $0\neq xyz \in P$, then either $xy \in P$ or $z \in P$. In addition to give many properties and characterizations of weakly 1-absorbing prime ideals, we also determine rings in which every proper ideal is weakly 1-absorbing prime. Furthermore, we investigate weakly 1-absorbing prime ideals in $C(X)$, which is the ring of continuous f…
We prove that for all $n$, simultaneously, we can choose prime filtrations of $R/I^n$ such that the set of primes appearing in these filtrations is finite.
Product of prime ideals as factorization of submodules
本文研究诺特环上模的子模的广义素理想分解，给出素理想乘积成为广义素理想分解的充要条件，并刻画素理想幂在分解中出现的条件。
Let $I$ and $J$ be two ideals of a commutative Noetherian ring $R$ and $M$ be an $R$-module of dimension $d$. If $R$ is a complete local ring and $M$ is finite, then attached prime ideals of $H^{d-1}_{I,J}(M)$ are computed by means of the concept of co-localization. Also, we illustrate the attached prime ideals of $H^{t}_{I,J}(M)$ on a non-local ring $R$, for $t= \dim M$ and $t= cd(I,J,M)$.
Let (T,m) be a complete local (Notherian) ring, C a finite set of pairwise incomparable nonmaximal prime ideals of T, and p a nonzero element. We provide necessary and sufficient conditions for T to be the completion of an integral domain A containing the prime ideal pA whose formal fiber is semilocal with maximal ideals the elements of C.
Associated primes of powers of edge ideals of edge-weighted trees
该研究针对递增加权树的边理想，完整刻画了其各次幂的伴随素数，丰富了交换代数中边理想幂的相关理论。
We exhibit the elementary but somewhat surprising property that most direct links of prime ideals in Gorenstein rings are equimultiple ideals. It leads to the construction of a bountiful set of Cohen--Macaulay Rees algebras.
In this paper, we introduce $ϕ$-1-absorbing prime ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity $1\neq0$ and $ϕ:\mathcal{I}(R)\rightarrow\mathcal{I}(R)\cup\{\emptyset\}$ be a function where $\mathcal{I}(R)$ is the set of all ideals of $R$. A proper ideal $I$ of $R$ is called a $ϕ$-1-absorbing prime ideal if for each nonunits $x,y,z\in R$ with $xyz\in I-ϕ(I)$, then either $xy\in I$ or $z\in I$. In addition to give many properties and characterizations of $ϕ$-1-absorbing prime ideals, we also determine rings in which every proper ideal is $ϕ$-1-absorbing prim…
It is well known that a domain without proper strongly divisorial ideals is completely integrally closed. In this paper we show that a domain without {\em prime} strongly divisorial ideals is not necessarily completely integrally closed, although this property holds under some additional assumptions.
We show that the Stanley's Conjecture holds for an intersection of four monomial prime ideals of a polynomial algebra $S$ over a field and for an arbitrary intersection of monomial prime ideals $(P_i)_{i\in [s]}$ of $S$ such that $P_i\not\subset Σ_{1=j\not =i}^s P_j$ for all $i\in [s]$.
In this paper, we give an affirmative answer to a conjecture raised by Polini and Ulrich.
We give an elementary proof that for a ring homomorphism A -> B, satisfying the property that every ideal in A is contracted from B, the following property holds: for every chain of prime ideals p_0 \subset ... \subset p_r in A there exists a chain of prime ideals q_0 \subset ... \subset q_r in B such that q_i \cap A = p_i.
Let $R$ be a Noetherian ring. We prove that $R$ has global dimension at most two if, and only if, every prime ideal of $R$ is of linear type. Similarly, we show that $R$ has global dimension at most three if, and only if, every prime ideal of $R$ is syzygetic. As a consequence, one derives a characterization of these rings using the André-Quillen homology.
For which monomial supports do most polynomials generate a prime ideal? We give necessary and sufficient conditions for the radical of the ideal to be prime over an algebraically closed field. In characteristic zero, the same conditions give primeness. As an application we show that under the same combinatorial conditions on Newton polytopes, the stable intersection of tropical hypersurfaces is connected through codimension one.
In this note, we show that a part of [5, Remark 2.2] is not correct. Some conditions are given under which the same holds.
Aghapournahr and Melkersson introduced the notion of Melkersson condition on a Serre subcategory of the module category over a commutative noetherian ring. This paper investigates the structure of set of prime ideals satisfying a Melkersson condition on a Serre subcategory. We try to calculate members of a set of these prime ideals for a subcategory consisting of extension modules in two given Serre subcategories of the module category. Meanwhile, we classify the structure of set of prime ideals satisfying a Melkersson condition over a 0-dimensional ring, a 1-dimensional local ring, and a 2-di…
Ideals as generalized prime ideal factorization of submodules
本文研究R-模子模的广义素理想分解问题，针对任意素理想乘积和给定模，给出存在以该乘积为广义素理想分解的子模的判定条件。
We present combinatorial characterizations for the associated primes of the second power of squarefree monomial ideals and criteria for this power to have positive depth or depth greater than one.
Let $G$ be a group, $R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set of all graded ideals of $R$. Suppose that $ϕ:GI(R)\rightarrow GI(R)\cup\{\emptyset\}$ is a function. In this article, we introduce and study the concept of graded $ϕ$-$1$-absorbing prime ideals. A proper graded ideal $I$ of $R$ is called a graded $ϕ$% -$1$-absorbing prime ideal of $R$ if whenever $a,b,c$ are homogeneous nonunit elements of $R$ such that $abc\in I-ϕ(I)$, then $ab\in I$ or $c\in I$. Several properties of graded $ϕ$-$1$-absorbing prime ideals have been examined.
We consider determinantal ideals, where the generating minors are encoded in a hypergraph. We study when the generating minors form a Gröbner basis. In this case, the ideal is radical, and we can describe algebraic and numerical invariants of these ideals in terms of combinatorial data of their hypergraphs, such as the clique decomposition. In particular, we can construct a minimal free resolution as a tensor product of the minimal free resolution of their cliques. For several classes of hypergraphs we find a combinatorial description of the minimal primes in terms of a prime splitting. That i…
Let $R$ be a Noetherian local ring. We prove that $R$ is regular of dimension at most four if, and only if, every prime ideal, defining a Gorenstein quotient ring, is syzygetic. We deduce a characterization of these rings in terms of the André-Quillen homology.
On the existence of the maximal ideal in the set of associated primes of monomial ideals
本文研究单项式理想幂的相伴素的性质，建立K[x,y,z]中极大理想是否属于其相伴素集的准则，证明n≥3时存在无穷多理想的相伴素集波动，并构造不满足持久性等性质的理想例子。
Let $B$ be a reduced local (Noetherian) ring with maximal ideal $M$. Suppose that $B$ contains the rationals, $B/M$ is uncountable and $|B| = |B/M|$. Let the minimal prime ideals of $B$ be partitioned into $m \geq 1$ subcollections $C_1, \ldots ,C_m$. We show that there is a reduced local ring $S \subseteq B$ with maximal ideal $S \cap M$ such that the completion of $S$ with respect to its maximal ideal is isomorphic to the completion of $B$ with respect to its maximal ideal and such that, if $P$ and $Q$ are prime ideals of $B$, then $P \cap S = Q \cap S$ if and only if $P$ and $Q$ are in $C_i…
Nil-prime ideals of a commutative ring
本文在交换环中引入幂零素理想概念以推广素理想，并研究幂零极大理想、幂零极小理想、幂零主理想等概念及其相应性质。
We show that, given integers $n_1,n_2, \ldots ,n_k$ with $2 < n_1 < n_2 < \cdots < n_k$, there exists a local (Noetherian) unique factorization domain that has maximal chains of prime ideals of lengths $n_1, n_2, \ldots ,n_k$ which are disjoint except at their minimal and maximal elements. In addition, we demonstrate that unique factorization domains can have other unusual prime ideal structures.
We show that all monomial ideals in the polynomial ring in at most 3 variables are pretty clean and that an arbitrary monomial ideal $I$ is pretty clean if and only if its polarization $I^p$ is clean. This yields a new characterization of pretty clean monomial ideals in terms of the arithmetic degree, and it also implies that a multicomplex is shellable if and only the simplicial complex corresponding to its polarization is (non-pure) shellable. We also discuss Stanley decompositions in relation to prime filtrations.
We find an explicit expression of the associated primes of monomial ideals as a colon by an element $v$, using the unique irredundant irreducible decomposition whose irreducible components are monomial ideals (Theorem 3.1). An algorithm to compute $v$ is given using Macaulay2 (Section 7). For squarefree monomial ideals the problem is related to the combinatorics of the underlying clutter or graph (Proposition 4.3). For ideals of Borel type the monomial $f$ takes a simpler form (Proposition 5.2). The authors classify when $f$ is unique (Proposition 6.2).
In a previous paper we exhibited the somewhat surprising property that most direct links of prime ideals in Gorenstein rings are equimultiple ideals with reduction number $1$. This led to the construction of large families of Cohen--Macaulay Rees algebras. The first goal of this paper is to extend this result to arbitrary Cohen--Macaulay rings. The means of the proof are changed since one cannot depend so heavily on linkage theory. We then study the structure of the Rees algebra of these links, more specifically we describe their canonical module in sufficient detail to be able to characterize…
We describe prime ideals of height 2 minimally generated by 3 elements in a Gorenstein, Nagata local ring of Krull dimension 3 and multiplicity at most 3. This subject is related to a conjecture of Y. Shimoda and to a long-standing problem of J. Sally.
The support of any module over a commutative ring is defined as the collection of all prime ideals of the ring at which the localization of the module is non-zero. For finitely generated modules, the support is the collection of all prime ideals containing the annihilator of the module. In this article, we raise the natural question that over which commutative rings, the support of every module is the collection of all the prime ideals of its annihilator. We completely classify such rings, and in the process it also comes out that it is enough to require that only for countably generated modul…
We study when $R \to S$ has the property that prime ideals of $R$ extend to prime ideals or the unit ideal of $S$, and the situation where this property continues to hold after adjoining the same indeterminates to both rings. We prove that if $R$ is reduced, every maximal ideal of $R$ contains only finitely many minimal primes of $R$, and prime ideals of $R[X_1,\dots,X_n]$ extend to prime ideals of $S[X_1,\dots,X_n]$ for all $n$, then $S$ is flat over $R$. We give a counterexample to flatness over a reduced quasilocal ring $R$ with infinitely many minimal primes by constructing a non-flat $R$-…
Gluing Associated Prime Ideals of Small Height
本文研究了局部环的相伴素理想粘合问题，提出了存在具有相同完备化且相伴素理想个数减少1的子环的充分条件，并给出了相关应用与推论。
Fluctuations in depth and associated primes of powers of ideals
本文研究了单项式理想BHH(m,r,s)的幂的深度函数特性，揭示其周期性行为及变量优化优势。
Let (R,m) be a Noetherian local domain of dimension n that is essentially finitely generated over a field and let R^ denote the m-adic completion of R. Matsumura has shown that n-1 is the maximal height possible for prime ideals of R^ in the generic formal fiber of R. In this article we prove that every prime ideal of R^ that is maximal in the generic formal fiber of R has height n-1. We also present a related result concerning the generic formal fibers of certain extensions of mixed polynomial-power series rings.
For the edge ideal I of an arbitrary simple graph G we describe the monomials of the saturation of a power of I in terms of (vertex) weighted graphs associated with the monomials. This description allows us to characterize the embedded associated primes of a power of I as covers of G which contain certain types of subgraphs of G. As an application, we completely classify the associated primes of the second and the third power of I in terms of G.
Let $T$ be a complete equicharacteristic local (Noetherian) UFD of dimension $3$ or greater. Assuming that $|T| = |T/m|$, where $m$ is the maximal ideal of $T$, we construct a local UFD $A$ whose completion is $T$ and whose formal fibers at height one prime ideals have prescribed dimension between zero and the dimension of the generic formal fiber. If, in addition, $T$ is regular and has characteristic zero, we can construct $A$ to be excellent.
Criteria for the presence of the maximal ideal in the set of associated primes
本文建立了多项式环中单项式理想幂的相伴素理想集中存在极大理想 (x1,...,xn) 的判别准则，并通过例子说明其适用性。
We use the method of characteristic sets with respect to two term orderings to prove the existence and obtain a method of computation of a bivariate Kolchin-type dimension polynomial associated with a non-reflexive difference-differential ideal in the algebra of difference-differential polynomials with several basic derivations and one translation. In particular, we obtain a new proof and a method of computation of the dimension polynomial of a non-reflexive prime difference ideal in the algebra of difference polynomials over an ordinary difference field. As a consequence, it is shown that the…
We continue the analysis of prime and semiprime operations over one-dimensional domains started in \cite{Va}. We first show that there are no bounded semiprime operations on the set of fractional ideals of a one-dimensional domain. We then prove the only prime operation is the identity on the set of ideals in semigroup rings where the ideals are minimally generated by two or fewer elements. This is not likely the case in semigroup rings with ideals of three or more generators since we are able to exhibit that there is a non-identity prime operations on the set of ideals of $k[[t^3,t^4,t^5]]$.
To compute the local cohomology of powers of edge ideals one needs to know their saturations. The saturation of the second and third powers has been described in terms of the graph in [13] and [10]. In this article, we give a combinatorial description of the generators of the saturation of the fourth power. As a consequence, we are able to give a complete classification of the associated primes of the fourth power of edge ideals in terms of the graph.
In this paper we study the set of prime ideals in vector lattices and how the properties of the prime ideals structure the vector lattice in question. The different properties that will be considered are firstly, that all or none of the prime ideals are order dense, secondly, that there are only finitely many prime ideals, thirdly, that every prime ideal is principal, and lastly, that every ascending chain of prime ideals is stationary (a property that we refer to as prime Noetherian). We also completely characterize the prime ideals in vector lattices of piecewise polynomials, which turns out…
Let R be a commutative ring with unity $1\in R$. In this article, we introduce the concept of prime principal right ideal rings (\textbf{PPRIR}), A prime ideal P of R is said to be prime principal right ideal (\textbf{PPRI}) is given by $P =\{ ar : r\in R\}$ for some element a. The ring R is said to be prime principal right ideal ring (\textbf{PPRIR}) if every prime ideal of R is a prime principal right ideal (\textbf{PPRI}). A prime principal right ideal ring R is called a prime principal right ideal domain (\textbf{PPRID}) if R is a domain. Several properties and characteristics of prime pri…
We characterize monomial ideals which are intersections of monomial prime ideals and study classes of ideals with this property, among them polymatroidal ideals.
The associated prime ideals of powers of polymatroidal ideals are studied, including the stable set of associated prime ideals of this class of ideals. It is shown that polymatroidal ideals have the persistence property and for transversal polymatroids and polymatroidal ideals of Veronese type the index of stability and the stable set of associated ideals is determined explicitly.
Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a "Prime Ideal Principle" that gives a uniform method of proving such facts, generalizing the Prime Ideal Principle for commutative rings due to T.Y. Lam and the author. Old and new "maximal implies prime" results are presented, with results touching on annihilator ideals, polynomial identity rings, the Artin-Rees property, Dedekind-finite rings, principal ideals generated by normal elements, strongly noetherian algebras, and just infinite algebras.
Generalizations of almost prime and right $S$-prime ideals in noncommutative rings
本文在非交换环中推广了几乎素理想概念，引入了几乎右 $S$-素理想，并利用 Nagata 理想化方法研究其在相关环中的性质与构造。
In the present paper, we investigate the commutativity of quotient ring $R/P$ where $R$ is any ring and $P$ is a prime ideal of $R$ which admits generalized derivations are satisfying some algebraic identities acting on prime ideals $P$.
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