The Pointwise Characterizations of Fuzzy Ideals and Fuzzy Prime Ideals

In this paper, the auther programs and discusses the fuzzy subrings, fuzzy ideals generated by fuzzy points.

Generalised Chain Conditions,Prime Ideals,and Classes of Locally Finite Lie Algebras

A Noetherian(Artinian)Lie algebra satisfies the maximal(minimal)condition for ideals.Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras.We study conditions on prime ideals relating these properties.We prove that the radical of any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals,and an ideally finite Lie algebra is quasi-Noetherian if and only if it is quasi-Artinian.Both properties are equivalent to soluble-by-finite.We also prove a structure theorem for serially finite Artinian Lie algebras.

Spectrum of L-fuzzy Prime Ideals of a Distributive Lattice

A topological space denoted by Fspec(D),called L-fuzzy prime spectrum of a bounded distributive lattice D is introduced.This space Fspec(D) is compact and it contains a subspace homeomorphic with the prime spectrum of D which is dense in it.The correspondence associating D to the topological space Fspec(D) is shown to define a contravarient functor from the category of bounded distributive lattices into the category of compact topological spaces.

EXPLICIT DESCRIPTION OF A CLASS OF INDECOMPOSABLE INJECTIVE MODULES

Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module R/p.

Ass(M/N)and Ann(M/N)on the Noetherian Module M

Let R is a noetherian ring,M is a finitely generated R-module.This paper studies the relation between associated prime Ass(M/N)and annihilator Ann(M/N),and has given the necessary and sufficient conditions of Ass(M/N)=Ann(M/N).

On a Characterization of Quasi-valuation Rings

Let R be a commutative ring without nil-factor. In this paper, we discuss the problem of quasi-valuation ring presented in the reference 'Wang Shianghaw, On quasi-valuation ring, Northeast People's Univ. Natur. Sci. J., (1)(1957), 27-40', when the quotient field of R is an algebraic number field or an algebraic function field, and we obtain a characterization of quasi-valuation rings.

On Weakly δ-Semiprimary Ideals of Commutative Rings

Let R be a commutative ring with 1≠ 0. A proper ideal I of R is a semiprimary ideal of R if whenever a, b ∈ R and a b ∈ I, we have a ∈ √I or b ∈√I; and I is a weakly semiprimary ideal of R if whenever a, b ∈ R and 0 ≠ ab ∈ √I, we have a ∈√I or b ∈ √I. In this paper, we introduce a new class of ideals that is closely related to the class of （weakly） semiprimary ideals. Let I（R） be the set of all ideals of R and let δ ： I（R） → I（R） be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J I, we have L δ（L） and δ（J） δ（I）. Let δ be an expansion function of ideals of R. Then a proper ideal I of R is called a δ-semiprimary （weakly δ-semiprimary） ideal of R if ab ∈ I （0 ≠ ab ∈ I） implies a ∈ δ（I） or b∈ δ（I）. A number of results concerning weakly δ-semiprimary ideals and examples of weakly δ-semiprimary ideals are given.

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 Grobner bases directly.

A CRITERION FOR TESTING WHETHER A DIFFERENCE IDEAL IS PRIME

This paper presents a criterion for testing the irreducibility of a polynomial over an algebraicextension field.Using this criterion and the characteristic set method,the authors give a criterion fortesting whether certain difference ascending chains are strong irreducible,and as a consequence,whetherthe saturation ideals of these ascending chains are prime ideals.

On 2-Absorbing Quasi-Primary Ideals in Commutative Rings

Let R be a commutative ring with nonzero identity.In this article,weintroduce the notion of 2-absorbing quasi-primary ideal which is a generalization ofquasi-primary ideal.We define a proper ideal I of R to be 2-absorbing quasi primaryif√1 is a 2-absorbing ideal of R.A number of results concerning 2-absorbing quasi-primary ideals and examples of 2-absorbing quasi-primary ideals are given.

The Heisenberg double of the quantum Euclidean group and its representations

The Heisenberg double D_(q)(E_(2))of the quantum Euclidean group O_(q)(E_(2))is the smash product of O_(q)(E_(2))with its Hopf dual U_(q)(e_(2)).For the algebra D_(q)(E_(2)),explicit descriptions of its prime,primitive and maximal spectra are obtained.All the prime factors of D_(q)(E_(2))are presented as generalized Weyl algebras.As a result,we obtain that the algebra D_(q)(E_(2))has no finite-dimensional representations,and D_(q)(E_(2))cannot have a Hopf algebra structure.The automorphism groups of the quantum Euclidean group and its Heisenberg double are determined.Some centralizers are explicitly described via generators and defining relations.This enables us to give a classification of simple weight modules and the so-called a-weight modules over the algebra D_(q)(E_(2)).

Higher Level Orderings on Modules

The aim of this paper is to investigate higher level orderings on modulesover commutative rings. On the basis of the theory of higher level orderings on fields andcommutative rings, some results involving existence of higher level orderings are generalized to thecategory of modules over commutative rings. Moreover, a strict intersection theorem for higherlevel orderings on modules is established.

S-fuzzy Version of Stone's Theorem for Distributive Lattices

In this paper,we initiate a study of S-fuzzy ideal(filter) of a lattice where S stands for a meet semilattice.A S-fuzzy prime ideal(filter) of a lattice is defined and it is proved that a S-fuzzy ideal(filter) of a lattice is S-fuzzy prime ideal(filter) if and only if any non-empty α-cut of it is a prime ideal(filter).Stone's theorem for a distributive lattice is extended by considering S-fuzzy ideals(filters).