Quotient rings induced via fuzzy ideals

This note we give a construction of a quotient ringR/μ induced via a fuzzy idealμ ina ringR. The Fuzzy First, Second and Third Isomorphism Theorems are established. For some applications of this construction of quotient rings, we show that ifμ is a fuzzy ideal of a commutative ringR, thenμ is prime (resp. maximal, primary) if and only ifR/μ is an integral domain (resp.R/μ is a field, every zero divisor inR/μ is nilpotent). Moreover we give a simpler characterization of fuzzy maximal ideal of a ring.     

Topological prime radical of a group

In this paper, we consider two approaches toward the definition of a topological prime radical of a topological group. In the first approach, the prime quasi-radical η(G) is defined as the intersection of all closed prime normal subgroups of a topological group G. Its properties are investigated. In the second approach, we consider the set η′(G) of all topologically strictly Engel elements of a topological group G. Its properties are investigated. It is proved that η′(G) is a radical in the class of all topological groups possessing a basis of neighborhoods of the identity element consisting of normal subgroups. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 4, pp. 15–22, 2004.     

On some equations in prime rings

The main purpose of this paper is to prove the following result. Let R be a prime ring of characteristic different from two and let T : R → R be an additive mapping satisfying the relation T(x ³) = T(x)x ² − xT(x)x + x ² T(x) for all x ∈ R. In this case T is of the form 4T(x) = qx + xq, where q is some fixed element from the symmetric Martindale ring of quotients. This result makes it possible to solve some functional equations in prime rings with involution which are related to bicircular projections.     

ζ μ -sets in generalized topological spaces

We define ζ _μ -sets, (ζ,μ)-closed sets and generalized ζ _μ -sets in a generalized topological space and investigate properties of several low separation axioms of generalized topologies constructed by the families of these sets. Characterizations of some properties of (ζ,μ)-R ₀ and (ζ,μ)-R ₁ generalized topological spaces will be given.     

Further Results on Finitely Generated Projective Modules

In this paper, the exchange rings R whose primitive factor rings are artinian are studied. The following results are proved: for any exchange ring R and any two-sided ideal I of R, K ₀(π) : K ₀(R)→K ₀(R/I) is a group epimorphism with the kernel {[P]−[Q] |P = PI, Q = QI}; there is an isomorphism of ordered groups from K ₀(R) to the gorup of all such functions ƒ _P : X→Q(P∈p(R)), where X is the set of all primitive ideals of R and Q, the rational integers. Received February 2, 1999, Accepted December 9, 1999     

Weakly Special Classes of Semiprime Rings

This paper investigates closure properties possessed by certain classes of finite subdirect products of prime rings. If ℳ is a special class of prime rings then the class ℳ_∞ of all finite subdirect products of rings in ℳ is shown to be weakly special. A ring S is said to be a right tight extension [resp. tight extension] of a subring R if every nonzero right ideal [resp. right ideal and left ideal] of S meets R nontrivially. Every hereditary class of semiprime rings closed under tight extensions is weakly special. Each of the following conditions imposed on a semiprime ring yields a hereditary class closed under right tight extensions: ACC on right annihilators; finite right Goldie dimension; right Goldie. The class of all finite subdirect products of uniformly strongly prime rings is shown to be closed under tight extensions, answering a published question. This revised version was published online in June 2006 with corrections to the Cover Date.     

Homological properties of the perfect and absolute integral closures of Noetherian domains

For a Noetherian local domain R let R ⁺ be the absolute integral closure of R and let R _∞ be the perfect closure of R, when R has prime characteristic. In this paper we investigate the projective dimension of residue rings of certain ideals of R ⁺ and R _∞. In particular, we show that any prime ideal of R _∞ has a bounded free resolution of countably generated free R _∞-modules. Also, we show that the analogue of this result is true for the maximal ideals of R ⁺, when R has residue prime characteristic. We compute global dimensions of R ⁺ and R _∞ in some cases. Some applications of these results are given.     

On the Zero-Divisor Graph of a Ring

Let R be a commutative ring with identity, Z(R) its set of zero-divisors, and Nil(R) its ideal of nilpotent elements. The zero-divisor graph of R is Γ(R) = Z(R)\{0}, with distinct vertices x and y adjacent if and only if xy = 0. In this article, we study Γ(R) for rings R with nonzero zero-divisors which satisfy certain divisibility conditions between elements of R or comparability conditions between ideals or prime ideals of R. These rings include chained rings, rings R whose prime ideals contained in Z(R) are linearly ordered, and rings R such that {0} ≠ Nil(R) ? zR for all z ∈ Z(R)\Nil(R).     

The embedding of line graphs associated to the zero-divisor graphs of commutative rings

Let R be a commutative ring with identity and denote Γ(R) for its zero-divisor graph. In this paper, we study the minimal embedding of the line graph associated to Γ(R), denoted by L(Γ(R)), into compact surfaces (orientable or non-orientable) and completely classify all finite commutative rings R such that the line graphs associated to their zero-divisor graphs have genera or crosscaps up to two.     

Generalized APP-Rings

We say a ring R is (centrally) generalized left annihilator of principal ideal is pure (APP) if the left annihilator ? _R (Ra) ⁿ is (centrally) right s-unital for every element a ∈ R and some positive integer n. The class of generalized left APP-rings includes generalized left (principally) quasi-Baer rings and left APP-rings (and hence left p.q.-Baer rings, right p.q.-Baer rings, and right PP-rings). The class of centrally generalized left APP-rings is closed under finite direct products, full matrix rings, and Morita invariance. The behavior of the (centrally) generalized left APP condition is investigated with respect to various constructions and extensions, and it is used to generalize many results on generalized PP-rings with IFP and semiprime left APP-rings. Moreover, we extend a theorem of Kist for commutative PP rings to centrally generalized left APP rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. Furthermore, we give a complete characterization of a considerably large family of centrally generalized left APP rings which have a sheaf representation.     

A local variational relation and applications

In [BGH] the authors show that for a given topological dynamical system (X,T) and an open coveru there is an invariant measure μ such that infh _μ(T,ℙ)≥h _top(T,U) where infimum is taken over all partitions finer thanu. We prove in this paper that if μ is an invariant measure andh _μ(T,ℙ) > 0 for each ℙ finer thanu, then infh _μ(T,ℙ > 0 andh _top(T,U) > 0. The results are applied to study the topological analogue of the Kolmogorov system in ergodic theory, namely uniform positive entropy (u.p.e.) of ordern (n≥2) or u.p.e. of all orders. We show that for eachn≥2 the set of all topological entropyn-tuples is the union of the set of entropyn-tuples for an invariant measure over all invariant measures. Characterizations of positive entropy, u.p.e. of ordern and u.p.e. of all orders are obtained. We could answer several open questions concerning the nature of u.p.e. and c.p.e.. Particularly, we show that u.p.e. of ordern does not imply u.p.e. of ordern+1 for eachn≥2. Applying the methods and results obtained in the paper, we show that u.p.e. (of order 2) system is weakly disjoint from all transitive systems, and the product of u.p.e. of ordern (resp. of all orders) systems is again u.p.e. of ordern (resp. of all orders). Project supported by one hundred talents plan and 973 plan.     

Intersection theorems with geometric consequences

In this paper we prove that ifℱ is a family ofk-subsets of ann-set, μ₀, μ₁, ..., μ_s are distinct residues modp (p is a prime) such thatk ≡ μ₀ (modp) and forF ≠ F′ ≠ℱ we have |F ∩F′| ≡ μ_i (modp) for somei, 1 ≦i≦s, then |ℱ|≦( _s ⁿ ). As a consequence we show that ifR ⁿ is covered bym sets withm<(1+o(1)) (1.2) ⁿ then there is one set within which all the distances are realised. It is left open whether the same conclusion holds for compositep.     

Integral inequalities with weights for maxiamal operators

Let μ be a measure on the upper half-space R ₊ ⁿ⁺¹ , and v a weight onR ⁿ, we give a characterization for the pair (v, μ) such that ∥M(fv)∥_{L Θ (μ}) ⩽ c ∥f∥_{L Θ (μ}), where Φ is an N-function satisfying Δ₂ condition andMf(x,t), is the maximal function onR ₊ ⁿ⁺¹ , which was introduced by Ruiz, F. and Torrea, J.. Supported by NSFC.     

Higher order Riesz transforms associated with Bessel operators

In this paper we investigate Riesz transforms R _μ ^(k) of order k≥1 related to the Bessel operator Δ_μ f(x)=-f”(x)-((2μ+1)/x)f’(x) and extend the results of Muckenhoupt and Stein for the conjugate Hankel transform (a Riesz transform of order one). We obtain that for every k≥1, R _μ ^(k) is a principal value operator of strong type (p,p), p∈(1,∞), and weak type (1,1) with respect to the measure dλ(x)=x ^2μ+1  dx in (0,∞). We also characterize the class of weights ω on (0,∞) for which R _μ ^(k) maps L ^p (ω) into itself and L ¹(ω) into L ^1,∞(ω) boundedly. This class of weights is wider than the Muckenhoupt class of weights for the doubling measure dλ. These weighted results extend the ones obtained by Andersen and Kerman.     

A note on the acyclicity of the Koszul complex of a module

We prove the vanishing of the Koszul homology group H _μ(Kos(M)_μ), where μ is the minimal number of generators of M. We give a counterexample that the Koszul complex of a module is not always acyclic and show its relationship with the homology of commutative rings.     

The structure of injective covers of special modules

In this paper, we prove that the injective cover of theR-moduleE(R/B)/R/B for a prime ideal B ofR is the direct sum of copies ofE(R/B) for prime ideals D ⊃ B, and if B is maximal, the injective cover is a finite sum of copies ofE(R/B). For a finitely generatedR-moduleM withn generators andG an injectiveR-module, we argue that the natural mapG ⁿ →G ⁿ/Hom _R (M, G) is an injective precover if Ext _R ¹ (M, R) = 0, and that the converse holds ifG is an injective cogenerator ofR. Consequently, for a maximal ideal R ofR, depth_R R ≧ 2 if and only if the natural mapE(R/R) →E(R/R)/R/R is an injective cover and depth_R R > 0.     

Weights andL Φ-boundedness of the Poisson integral operator

In this paper, we get a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL _Φ(R ⁿ, v(x)dx) to weak-L _Φ(R ₊ ⁿ⁺¹ ,dμ), where Φ is anN-function satisfying the Δ₂-condition. We also find a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL _Φ(R ⁿ,v(x)dx) toL _Φ(R ₊ ⁿ⁺¹ ,dμ) under some additional condition. Partially supported by NNSF of P.R. China     

Cyclotomic units over finite fields

Let ζ be a primitivesp-th root of unity for a primep>2, and consider the group Ω(ζ) of cyclotomic units in the ringR(ζ)=ℒ[ζ+ζ^-1]. This paper deals with the image of Ω(ζ) in the unit group ofR(ζ)/qR(ζ), whereq is a prime ≠p. In particular, it obtains criteria for this image to be essentially everything, and a lower bound on the density of primesp (withq fixed) for which it cannot be. These results have a direct bearing on previous work about units in integral group rings for cyclic groups of orderpq. Work supported in part by an operating grant from NSERC (Canada).     

FGT-injective dimensions of Π-coherent rings and almost excellent extension

We study, in this article, the FGT-injective dimensions of Π-coherent rings. If R is right Π-coherent, and TI (resp. TF) stands for the class of FGT-injective (resp. FGT-flat) R-modules (n ≥ 0), we show that the following are equivalent: