On the stable set of associated prime ideals of a monomial ideal

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.     

Monomial localizations and polymatroidal ideals

In this paper we consider monomial localizations of monomial ideals and conjecture that a monomial ideal is polymatroidal if and only if all its monomial localizations have a linear resolution. The conjecture is proved for squarefree monomial ideals where it is equivalent to a well-known characterization of matroids. We prove our conjecture in many other special cases. We also introduce the concept of componentwise polymatroidal ideals and extend several of the results known for polymatroidal ideals to this new class of ideals.     

The Extensional Structure of Commutative Noetherian Rings

Abstract

We show that finitely generated modules over a commutative Noetherian ring can be classified, up to isomorphism of submodule series, in a manner analogous to the classification of integers as products of prime numbers. In outline, two such modules have isomorphic submodule series if and only if 1) the set of minimal associated prime ideals of these modules coincide, 2) the multiplicities of these modules at these prime ideals coincide, and 3) the modules represent the same element in a certain group corresponding to the above set of prime ideals. Regarding condition 3), we show that, in the very special case that the ring is a Dedekind domain, the group corresponding to the prime ideal (0) is the ideal class group of the ring.     

Polynomes a valeurs entieres sur un anneau de pseudovaluation

We study the ring of integral valued polynomials over a pseudovaluation domain A. We entirely determine the set of prime ideals above the maximal ideal M of A: if M is a principal ideal in the valuation domain V associated with A and if its residue field is finite, then this set is in bijection with a topologically complete ring, as in the Noetherian case; if M is principal but of infinite residue field in V, then this set is finite; at last, if M is not principal, then the ring of integral valued polynomials is included in V[X] and has the same set of prime ideals above M.     

On the Stanley depth of weakly polymatroidal ideals

Let ${\mathbb{K}}$ be a field and ${S = \mathbb{K}[x_1,\dots,x_n]}$ be the polynomial ring in n variables over the field ${\mathbb{K}}$ . In this paper, it is shown that Stanley’s conjecture holds for I and S/I if I is a product of monomial prime ideals or I is a high enough power of a polymatroidal or a stable ideal generated in a single degree.     

Semiprime Ideals and Separation Theorems for Posets

The concept of a semiprime ideal in a poset is introduced. The relations between the semiprime (prime) ideals of a poset and the ideals of the set of all ideals of the poset are established. A result analogous to Separation Theorem is obtained in respect of semiprime ideals. Further, a generalization of Stone’s Separation Theorem for posets is obtained in respect of prime ideals. Some counterexamples are also given.      

一种几何定理机器证明的零维化方法

本文对一类初等几何定理的证明给出了一种机械化方法，利用这种方法，可计算出一个由有限个素理想组成的集合，所有属于假设部分对应的某一扩域上的理想的素理想都在这个集合中出现并且可以挑选出来．因而一个几何定理一般真确，当且仅当终结多项式属于全部的这种素理想，即对其不可约特征列的余式为零．     

关于偏序集相关联图的色数

对于每一个含有最小元素0的偏序集（P,≤）可以得到一个与其关联的图G（P）.本文主要通过代数的方法研究了所得关联图G（P）的性质,证明了如果G（P）的色数和团数是有限的,那么色数和团数都仅比P的极小素理想的个数大1.     

Linked Primes and Orders in Artinian Rings

It is shown that prime ideals of a Noetherian ring are linked if and only if certain corresponding prime ideals are linked in an associated Artinian ring. Furthermore, it is shown that there is a canonical linking ideal, which can be found by using a construction based on middle annihilator ideals.     

Uppers to zero in intersections of prime ideals

Let R be a Noetherian integral domain. The structure of the partially-ordered set of prime ideals of R[z], the polynomial ring in one indeterminate over R, is not fully understood. I demonstrate that if p₁,…,p_n are prime ideals in R[x] with ht(p_i) > 2 and either n = 1 or R is not a Henselian local domain of dimension < 2, then pi D-o-C\pn contains [R] many prime ideals which intersect R at (0). I also show that if R is a Noetherian domain that is not a Henselian local domain and p₁,…,p_n are prime ideals with height > 2 each of which contains a monic polynomial, then their intersection contains [R] many prime ideals meeting R at (0), each containing a monic polynomial.     

SEMIGROUPS CHARACTERIZED BY THEIR FUZZY IDEALS

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Space of minimal prime ideals of a ring without nilpotent elements

Hull-kernel topology on the set ∑(R) of prime ideals of a ring R with unity and without nilpotent elements is discussed. The restriction of this topology to the set π(R) of minimal prime ideals of R has been investigated in detail. The compactness of π(R) has been characterized in several ways. An interesting characterization of Baer rings is given.A functorial correspondence between the category of rings having the property that every prime ideal contains a unique minimal prime ideal and their minimal spectra is established.     

Split Quaternions and Integer-valued Polynomials

The integer split quaternions form a noncommutative algebra over ?. We describe the prime and maximal spectrum of the integer split quaternions and investigate integer-valued polynomials over this ring. We prove that the set of such polynomials forms a ring, and proceed to study its prime and maximal ideals. In particular we completely classify the primes above 0, we obtain partial characterizations of primes above odd prime integers, and we give sufficient conditions for building maximal ideals above 2.     

A going-up theorem for arbitrary chains of prime ideals

We prove that if an extension R ? T of commutative rings satisfies the going-up property (for instance, if T is an integral extension of R), then any increasing chain of prime ideals of R (indexed by an arbitrary linearly ordered set) is covered by some corresponding chain of prime ideals of T. As a corollary, we recover the recent result of Kang and Oh that any such chain of prime ideals of an integral domain D is covered by a corresponding chain in some valuation overring of D.     

Differential Calculi on Quantum Groupoids

Let R be a commutative ring with Noetherian spectrum in which zero is a primary ideal. We determine the minimal zero-dimensional extensions of R when every regular prime ideal of R is contained in only finitely many prime ideals. This extends previous results of the first author for dim (R) ≤1. We also present a characterization of the partially ordered set of prime ideals in a ring with Noetherian spectrum.     

Fuzzy semirings

In this paper we initiate the study of fuzzy semirings and fuzzy A-semimodules where A is a semiring and A-semimodules are representations of A. In particular, semirings all of whose ideals are idempotent, called fully idempotent semirings, are investigated in a fuzzy context. It is proved, among other results, that a semiring A is fully idempotent if and only if the lattice of fuzzy ideals of A is distributive under the sum and product of fuzzy ideals. It is also shown that the set of proper fuzzy prime ideals of a fully idempotent semiring A admits the structure of a topological space, called the fuzzy prime spectrum of A.     

Ideal Structure and Stable Rank of $${\mathbb {C} e+ \ell^2(I)}$$ with the Hadamard Product

Let I be any index set. We consider the Banach algebra \mathbb C e+ l²(I){\mathbb {C} e+ \ell^2(I)} with the Hadamard product, and prove that its Bass and topological stable ranks are both equal to 1. We also characterize divisors, maximal ideals, closed ideals and closed principal ideals. For I=\mathbb N{I=\mathbb {N}} we also characterize all prime z-ideals in this Banach algebra.     

Noncommutative Generalizations of Theorems of Cohen and Kaplansky

This paper investigates situations where a property of a ring can be tested on a set of “prime right ideals.” Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff every “prime right ideal” is finitely generated (resp. principal), where the phrase “prime right ideal” can be interpreted in one of many different ways. We also use our methods to show that other properties can be tested on special sets of right ideals, such as the right artinian property and various homological properties. Applying these methods, we prove the following noncommutative generalization of a result of Kaplansky: a (left and right) noetherian ring is a principal right ideal ring iff all of its maximal right ideals are principal. A counterexample shows that the left noetherian hypothesis cannot be dropped. Finally, we compare our results to earlier generalizations of Cohen’s and Kaplansky’s theorems in the literature.