Asymptotic behaviour of certain sets of associated prime ideals of Ext-modules

Let R be a commutative Noetherian ring, be an ideal of R and M be a finitely generated R-module. Melkersson and Schenzel asked whether the set becomes stable for a fixed integer i and sufficiently large j. This paper is concerned with this question. In fact, we prove that if s ≥ 0 and n ≥ 0 such that for all i with i < n, then is finite for all i with i < n, and is finite for all i with i ≤ n, where for a subset T of Spec(R), we set . Also, among other things, we show that if n ≥ 0, R is semi-local and is finite for all i with i < n, then is finite for all i with i ≤ n. K. Khashyarmanesh was partially supported by a grant from Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 86130027).     

A linear function associated to asymptotic prime divisors

Let be a Noetherian standard -graded ring and finitely generated, -graded -modules. Let be finitely many homogeneous ideals of . We show that there exist linear functions such that the associated primes over of and are stable whenever satisfies and , respectively.

     

Projective ideals of skew polynomial rings over HNP rings

Let R be an hereditary Noetherian prime ring (or, HNP-ring, for short), and let S?=?R[x;σ] be a skew polynomial ring over R with σ being an automorphism on R. The aim of the paper is to describe completely the structure of right projective ideals of R[x;σ] where R is an HNP-ring and to obtain that any right projective ideal of S is of the form X𝔟[x;σ], where X is an invertible ideal of S and 𝔟 is a σ-invariant eventually idempotent ideal of R.     

Noetherian rings of low global dimension and syzygetic prime ideals

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, we derive a characterization of these rings using the André-Quillen homology.     

Convergence of sequences of sets of associated primes

It is a well-known result of M. Brodmann that if is an ideal of a commutative Noetherian ring , then the set of associated primes of the -th power of is constant for all large . This paper is concerned with the following question: given a prime ideal of which is known to be in for all large integers , can one identify a term of the sequence beyond which will subsequently be an ever-present? This paper presents some results about convergence of sequences of sets of associated primes of graded components of finitely generated graded modules over a standard positively graded commutative Noetherian ring; those results are then applied to the above question.

     

When Essential Extensions of Finitely Generated Modules are Finitely Generated

For certain classes 𝒞 of R-modules, including singular modules or modules with locally Krull dimensions, it is investigated when every module in 𝒞 with a finitely generated essential submodule is finitely generated. In case 𝒞 = Mod-R, this means E(M)/M is Noetherian for any finitely generated module M_R. Rings R with latter property are studied and shown that they form a class 𝒬 properly between the class of pure semisimple rings and the class of certain max rings. Duo rings in 𝒬 are precisely Artinian rings. If R is a quasi continuous ring in 𝒬 then R ? A ⊕ T where A is a semisimple Artinian ring and T ∈ 𝒬 with Z(T_T) ≤_ess T_T.     

Average distribution of prime ideals in families of number fields

We view an algebraic curve over ℚ as providing a one-parameter family of number fields and obtain bounds for the average value of some standard prime ideal counting functions over these families which are better than averaging the standard estimates for these functions.      

Minimal prime ideals of reduced rings

Let R be a reduced ring with Q its Martindale symmetric ring of quotients, and let B be the complete Boolean algebra of all idempotents in C, where C is the extended centroid of R. It is proved that every minimal prime ideal of R must be of the form mQ∩R for some maximal ideal m of B but the converse is in general not true. In addition, if R is centrally closed or has only finitely many minimal prime ideals, then the converse also holds. By applying the explicit expression, many properties of minimal prime ideals of reduced rings are realized more easily.     

A bracket power characterization of analytic spread one ideals

The main theorem characterizes, in terms of bracket powers, analytic spread one ideals in local rings. Specifically, let be regular nonunits in a local (Noetherian) ring and assume that , the integral closure of , where . Then the main result shows that for all but finitely many units in that are non-congruent modulo and for all large integers and it holds that for and not divisible by , where is the -th bracket power of . And, conversely, if there exist positive integers , , and such that has a basis such that , then has analytic spread one.

     

Replication in critical graphs and the persistence of monomial ideals

Motivated by questions about square-free monomial ideals in polynomial rings, in 2010 Francisco et al. conjectured that for every positive integer k and every k-critical (i.e., critically k  -chromatic) graph, there is a set of vertices whose replication produces a (k+1)$(k+1)$-critical graph. (The replication of a set W of vertices of a graph is the operation that adds a copy of each vertex w in W, one at a time, and connects it to w and all its neighbours.)     

On noncommutative piecewise Noetherian rings

We extend the definition of a piecewise Noetherian ring to the noncommutative case, and investigate various properties of such rings. In particular, we show that a ring with Krull dimension is piecewise Noetherian. Certain fully bounded piecewise Noetherian rings have Gabriel dimension and exhibit the Gabriel correspondence between prime ideals and indecomposable injective modules.     

On the Hartshorne-Speiser-Lyubeznik theorem about Artinian modules with a Frobenius action

Let be a commutative Noetherian local ring of prime characteristic. The purpose of this paper is to provide a short proof of G. Lyubeznik's extension of a result of R. Hartshorne and R. Speiser about a module over the skew polynomial ring (associated to and the Frobenius homomorphism , in the indeterminate ) that is both -torsion and Artinian over .

     

Submonoids of Polycyclic-by-Finite Groups and their Algebras

We describe Noetherian semigroup algebras K[S] of submonoids S of polycyclic-by-finite groups over a field K. As an application, we show that these algebras are finitely presented and also that they are Jacobson rings. Next we show that every prime ideal P of K[S] is strongly related to a prime ideal of the group algebra of a subgroup of the quotient group of S via a generalised matrix ring structure on K[S]/P. Applications to the classical Krull dimension, prime spectrum, and irreducible K[S]-modules are given.     

The kernels of radical homomorphisms and intersections of prime ideals

We establish a necessary condition for a commutative Banach algebra so that there exists a homomorphism from into another Banach algebra such that the prime radical of the continuity ideal of is not a finite intersection of prime ideals in . We prove that the prime radical of the continuity ideal of an epimorphism from onto another Banach algebra (or of a derivation from into a Banach -bimodule) is always a finite intersection of prime ideals. Under an additional cardinality condition (and assuming the Continuum Hypothesis), this necessary condition is proved to be sufficient. En route, we give a general result on norming commutative semiprime algebras; extending the class of algebras known to be normable. We characterize those locally compact metrizable spaces for which there exists a homomorphism from into a radical Banach algebra whose kernel is not a finite intersection of prime ideals.

     

The decision of prime and primary ideal

We give more efficient criteria to characterise prime ideal or primary ideal. Further, we obtain the necessary and sufficient conditions that an ideal is prime or primary in real field from the Gröbner bases directly.