The primary purpose of this paper is give a classification scheme for the nonzero primes of a Prüfer domain based on five properties. A prime $P$ of a Prüfer domain $R$ could be sharp or not sharp, antesharp or not, divisorial or not, branched or unbranched, idempotent or not. Based on these five basic properties, there are six types of maximal ideals and twelve types of nonmaximal (nonzero) primes. Both characterizations and examples are given for each type that exists.
We give a description of the minimal primes of the ideal generated by the 2 x 2 adjacent minors of a generic matrix. We also compute the complete prime decomposition of the ideal of adjacent m x m minors of an m x n generic matrix when the characteristic of the ground field is zero. A key intermediate result is the proof that the ideals which appear as minimal primes are, in fact, prime ideals. This introduces a large new class of mixed determinantal ideals that are prime.
Using the theory of asymptotic test ideals, we prove the prime characteristic analogue of a characteristic $0$ result of Ein, Lazarsfeld and Smith (arXiv:math/0202303) on uniform approximation of valuation ideals associated to real-valued Abhyankar valuations centered on regular varieties over perfect fields.
Let $I$ be a monomial ideal of a polynomial ring $R$. In this paper we determine a number $B$ such that $\Ass (I^n/I^{n+1}) = \Ass (I^{B}/I^{B+1})$ for all $n\geq B$.
Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms of the associated primes of select powers of $I$ and of $J$. We record two applications. First, in case the field is algebraically closed, we construct primary decompositions for powers of $I+J$ from primary decompositions for powers of $I$ and $J$. Separately, we attack the persistence problem for associated primes of powers of an ideal in case one of $I$ or…
Let $f:A \to B$ be a ring homomorphism and let $J$ be an ideal of $B$. In this paper, we study the amalgamation of $A$ with $B$ along $J$ with respect to $f$ (denoted by ${A\Join^fJ}$), a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced and studied by D'Anna and Fontana in 2007, and other classical constructions (such as the $A+ XB[X]$, the $A+ XB[[X]]$ and the $D+M$ constructions). In particular, we completely describe the prime spectrum of the amalgamated duplication and we give bounds for its Krull dimension.
Suppose that $R$ is a local domain essentially of finite type over a field of characteristic 0, and $ν$ a valuation of the quotient field of $R$ which dominates $R$. The rank of such a valuation often increases upon extending the valuation to a valuation dominating $\hat R$, the completion of $R$. When the rank of $ν$ is 1, Cutkosky and Ghezzi handle this phenomenon by resolving the prime ideal of infinite value, but give an example showing that when the rank is greater than 1, there is no natural ideal in $\hat R$ that leads to this obstruction. We extend their result on the resolution of pri…
This paper contributes to the study of the prime spectrum and dimension theory of symbolic Rees algebra over Noetherian domains. We first establish some general results on the prime ideal structure of subalgebras of affine domains, which actually arise, in the Noetherian context, as domains between a domain $A$ and $A[a^{-1}]$. We then examine closely the special context of symbolic Rees algebras (which yielded the first counter-example to the Zariski-Hilbert problem). One of the results states that if $A$ is a Noetherian domain and $p$ a maximal ideal of $A$, then the Rees algebra of $p$ inhe…
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.
This paper is concerned with the prime spectrum of a tensor product of algebras over a field. It seeks necessary and sufficient conditions for such a tensor product to have the S-property, strong S-property, and catenarity. Its main results lead to new examples of stably strong S-rings and universally catenarian rings. The work begins by investigating the minimal prime ideal structure. Throughout, several results on polynomial rings are recovered, and numerous examples are provided to illustrate the scope and sharpness of the results.
Let S=K[x_1,...,x_n] be a polynomial ring. Denote by $p_a$ the power sum symmetric polynomial x_1^a+...+x_n^a. We consider the following two questions: Describe the subsets $A \subset \mathbb{N}$ such that the set of polynomials $p_a$ with $a \in A$ generate a prime ideal in S or the set of polynomials $p_a$ with $a \in A$ is a regular sequence in S. We produce a large families of prime ideals by exploiting Serre's criterion for normality [4, Theorem 18.15] with the help of arithmetic considerations, vanishing sums of roots of unity [9]. We also deduce several other results concerning regular …
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.
Let $G$ be a flat finite-type group scheme over a scheme $S$, and $X$ a noetherian $S$-scheme on which $G$-acts. We define and study $G$-prime and $G$-primary $G$-ideals on $X$ and study their basic properties. In particular, we prove the existence of minimal $G$-primary decomposition and the well-definedness of $G$-associated $G$-primes. We also prove a generalization of Matijevic-Roberts type theorem. In particular, we prove Matijevic-Roberts type theorem on graded rings for $F$-regular and $F$-rational properties.
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.
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…
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 characterize monomial ideals which are intersections of monomial prime ideals and study classes of ideals with this property, among them polymatroidal ideals.
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.
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.
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.
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 $i\in [s]$.
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.
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.
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…
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…
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 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.
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_…
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…
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.
