Holger Brenner
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.
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Lifting multiplicative lattices to ideal sytems
本文研究乘法格与理想系统的联系，给出一种将乘法格提升为幺半群上（弱）理想系统的机制。
In the article "Non-commutative Grobner bases for commutative algebras", Eisenbud-Peeva-Sturmfels proved a number of results regarding Grobner bases and initial ideals of those ideals in the free associative algebra which contain the commutator ideal. We prove similar results for ideals which contains the anti-commutator ideal (the defining ideal of the exterior algebra). We define one notion of generic initial ideals in the free assoicative algebra, and show that gin's of ideals containing the commutator ideal, or the anti-commutator ideal, are finitely generated.
In this paper, using the canonical correspondence between the idempotents and clopens, we obtain several new results on lifting idempotents. The Zariski clopens of the maximal spectrum are precisely determined, then as an application, lifting idempotents modulo the Jacobson radical is characterized. Lifting idempotents modulo an arbitrary ideal is also characterized in terms of certain connected sets related to that ideal. Then as an application, we obtain that the sum of a lifting ideal and a regular ideal is a lifting ideal. We prove that lifting idempotents preserves the orthogonality in co…
Computing Generic Fibers of Polynomial Ideals with FGLM and Hensel Lifting
本文提出结合FGLM算法与Hensel提升策略的新算法，用于计算正维多项式理想的通用纤维，复杂度与展开项数拟线性，实验验证其高效性。
We show how to lift any monomial ideal J in n variables to a saturated ideal I of the same codimension in n+t variables. We show that I has the same graded Betti numbers as J and we show how to obtain the matrices for the resolution of I. The cohomology of I is described. Making general choices for our lifting, we show that I is the ideal of a reduced union of linear varieties with singularities that are `as small as possible' given the cohomological constraints. The case where J is Artinian is the nicest. In the case of curves we obtain stick figures for I, and in the case of points we obtain…
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.
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.
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.
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.
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…
We study the defining equations of the Rees algebra of ideals arising from curve parametrizations in the plane and in rational normal scrolls, inspired by the work of Madsen and Kustin, Polini and Ulrich. The curves are related by work of Bernardi, Gimigliano and Ida, and we use this framework to relate the defining equations.
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.
Liftings of ideals in positive characteristic to those in characteristic zero:Surface case
本文引入了正特征到零特征的理想提升概念，通过骨架方法关联正特征与零特征的奇点不变量，并证明了光滑曲面与多理想对的log discrepancy集合离散，以及正特征中最小log discrepancy和log canonical阈值集合包含于零特征中。
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 study how to lift Markov bases and Gröbner bases along linear maps of lattices. We give a lifting algorithm that allows to compute such bases iteratively provided a certain associated semigroup is normal. Our main application is the toric fiber product of toric ideals, where lifting gives Markov bases of the factor ideals that satisfy the compatible projection property. We illustrate the technique by computing Markov bases of various infinite families of hierarchical models. The methodology also implies new finiteness results for iterated toric fiber products.
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 $P$.
Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of weakly $1$-absorbing prime ideals which is a generalization of weakly prime ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing prime if for all nonunit elements $a,b,c \in R$ such that $0\neq abc \in I$, then either $ab \in I$ or $c \in I$. A number of results concerning weakly $1$-absorbing prime ideals and examples of weakly $1$-absorbing prime ideals are given. It is proved that if $I$ is a weakly $1$-absorbing prime ideal of a ring $R$ and $0 \neq I_1I_2I_3 \subseteq I$ for some ideals $I_…
Prime and maximal ideals in Hurwitz polynomial rings
本文刻画了Hurwitz多项式环中R-不相交理想为素理想的条件，并研究了极大理想的存在性及其由最小次数多项式生成的判定方法。
In this paper, we study associated primes and depth of integral closures of powers of edge ideals. We provide sharp bounds on how big of powers for which the set of associated primes and the depth of integral closures of powers of edge ideals are stable.
Let M be a fixed left R-module. For a left R-module X, we introduce the notion of M-prime (resp. M-semiprime) submodule of X such that in the case M=R, which coincides with prime (resp. semiprime) submodule of X. Other concepts encountered in the general theory are M-m-system sets, M-n-system sets, M-prime radical and M-Baer's lower nilradical of modules. Relationships between these concepts and basic properties are established. In particular, we identify certain submodules of M, called "prime M-ideals", that play a role analogous to that of prime (two-sided) ideals in the ring R. Using this d…
We study the locus of the liftings of a homogeneous ideal $H$ in a polynomial ring over any field. We prove that this locus can be endowed with a structure of scheme $\mathrm L_H$ by applying the constructive methods of Gröbner bases, for any given term order. Indeed, this structure does not depend on the term order, since it can be defined as the scheme representing the functor of liftings of $H$. We also provide an explicit isomorphism between the schemes corresponding to two different term orders. Our approach allows to embed $\mathrm L_H$ in a Hilbert scheme as a locally closed subscheme, …
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