operations on fuzzy ideals in this paper we give suitable definitions of the operations of l fuzzy ideals in a ring as intersection addition multiplication and residual division and we prove some fundamental properties of the operations analogous to those that hold for the operations of ordinary ideals see 3

existence and uniqueness of fuzzy ideals let r be a commutative ring the idea of identifying a fuzzy ideal of r with its corresponding chain of level ideals and its range of values although widely used in the literature for the study of fuzzy ideals has never been investigated in a general setting before we investigate in this paper the more general problem of characterization of all those fuzzy ideals that can be identified with a given arbitrary family c of subsets of r together with a given arbitrary subset s of 0 1 necessary and sufficient conditions for the existence and uniqueness of such fuzzy ideals are established in particular we obtain that uniqueness always holds true for artinian rings moreover examples are presented to further support our results and shed more light on the above mentioned problem of identification we finish by raising a natural question that is left as an open problem for further investigation

fuzzy normal subgroups and fuzzy cosets abstract we introduce here the notion of 1 i a fuzzy normal subgroup and 2 ii a fuzzy coset we prove that for a group g a fuzzy subgroup is fuzzy normal if and only if it is constant on the conjugate classes of g further we show that the level subgroups of a fuzzy normal subgroup are all normal we obtain several results which are analogs of some basic theorems of group theory in particular a fuzzy version of lagrange s theorem is established

fuzzy lie ideals and fuzzy lie subalgebras in this paper we have tried to apply the concepts of fuzzy sets to lie algebra to define some concepts related to fuzzy sets and to discuss their properties and the relations among them

tl subrings and tl ideals part 1 basic concepts abstract in 25 27 we put forward a rudimentary frame of the theory of tl subalgebras in 26 we established a theory of tl subgroups and tl subgroups in this paper we will continue our work in concern with basic ideas provided in 25 27 introduce the notions of tl subrings and tl ideals of a ring and study some of their properties where t is an arbitrary infinitely distributive t norm on a given complete brouwerian lattice l

fuzzy nil radicals and fuzzy primary ideals abstract the purpose of this paper is to extend the concept of a fuzzy set by introducing the concept of a fuzzy nil radical of a fuzzy ideal in a ring using this concept a fuzzy primary ideal is defined some fundamental results concerning these notions are proved

twisted modules and g equivariantization in logarithmic conformal field theory a two dimensional chiral conformal field theory can be viewed mathematically as the representation theory of its chiral algebra a vertex operator algebra vertex operator algebras are especially well suited for studying logarithmic conformal field theory in which correlation functions have logarithmic singularities arising from non semisimple modules for the chiral algebra because of the logarithmic tensor category theory of huang lepowsky and zhang in this paper we study not necessarily semisimple or rigid braided tensor categories mathcal c of modules for the fixed point vertex operator subalgebra v g of a vertex operator super algebra v with finite automorphism group g the main results are that every v g module in mathcal c with a unital and associative v action is a direct sum of g twisted v modules for possibly several g in g that the category of all such twisted v modules is a braided g crossed super category and that the g equivariantization of this braided g crossed super category is braided tensor equivalent to the original category mathcal c of v g modules this generalizes results of kirillov and muger proved using rigidity and semisimplicity we also apply the main results to the orbifold rationality problem whether v g is strongly rational if v is strongly rational we show that v g is indeed strongly rational if v is strongly rational g is any finite automorphism group and v g is c 2 cofinite

on fuzzy ideals of a ring 1 abstract the aim of this paper is to study fuzzy ideals of a ring to characterise regular rings and noetherian rings by fuzzy ideals and to determine all fuzzy prime ideals of the ring z of integers

fuzzy ideals and fuzzy prime ideals of a ring abstract besides obtaining certain ring theoretic analogues some peculiarities of the fuzzy setting are also discussed in particular a study of i the conditions under which a given fuzzy ideal can or cannot be expressed as a union of two proper fuzzy ideals ii the algebraic nature of images and preimages of fuzzy prime ideals under homomorphisms and iii the ring of all the fuzzy cosets of a fuzzy prime fuzzy maximal ideal is carried out

normal fuzzy r subgroups in near rings in this paper we introduce the notion of a normal fuzzy r subgroup in a near ring and investigate some related properties

characterisation of fuzzy semiprime ideals of the ring of integers z abstract the aim of the paper is to characterise fuzzy semiprime ideals of the ring of integers z

fuzzy semiprimary ideals of rings abstract the concept of fuzzy ideals is extended by introducing fuzzy semiprimary ideals in rings this class of fuzzy ideals generalizes the class of fuzzy semiprimary ideals it is shown that there do exist fuzzy ideals which are fuzzy semiprimary but not fuzzy primary a study of i the level ideals of a fuzzy semiprimary ideal ii the characteristic function of a semiprimary ideal iii the zero divisors in the ring of all fuzzy cosets of a fuzzy semiprimary ideal and iv the algebraic nature of direct and inverse images of fuzzy semiprimary ideals under homomorphisms is carried out

level subgroups and union of fuzzy subgroups abstract in this paper we investigate the conditions under which a given fuzzy subgroup of a given group can or can not be realized as a union of two proper fuzzy subgroups moreover we formulate the notion of fuzzy subgroup generated by a given fuzzy subset by level subgroups we also give characterizations of fuzzy conjugate subgroups and fuzzy characteristic subgroups by their level subgroups also we discuss the relationship between level subgroups of a given fuzzy subgroup and the level subgroups of its homomorphic image

fuzzy prime ideals of rings abstract the notions of fuzzy ideal and fuzzy prime ideal of a ring with truth values in a complete lattice satisfying the infinite meet distributive law are introduced which generalize the existing notions with truth values in the unit interval of real numbers a procedure to construct any fuzzy ideal from a family of ideals with certain conditions is established all fuzzy prime maximal ideals of a given ring are determined by establishing a one to one correspondence between fuzzy prime maximal ideals and the pairs p where p is a prime maximal ideal of the ring and is a prime element dual atom in the complete lattice

fuzzy invariant subgroups and fuzzy ideals abstract the purpose of this paper is to introduce some basic concepts of fuzzy algebra as fuzzy invariant subgroups fuzzy ideals and to prove some fundamental properties in particular it will give a characteristic of a field by a fuzzy ideal

fuzzy semiprime ideals in semigroups abstract in this paper we shall introduce the notion of fuzzy semiprimality in a semigroup which is an extension of semiprimality in it and characterize a semigroup that is a semilattice of simple semigroups in terms of fuzzy semiprimality

fuzzy nil radicals and fuzzy semiprimary ideals of rings abstract an attempt is made to define a fuzzy semiprimary ideal by using the idea of the fuzzy nil radical of a fuzzy ideal the level ideals of a such a fuzzy ideal are examined and the ring of fuzzy cosets of such a fuzzy ideal is studied the algebraic nature of such fuzzy ideals under homomorphisms is also ascertained

pontryagin s principle for state constrained boundary control problems of semilinear parabolic equations this paper deals with state constrained optimal control problems governed by semilinear parabolic equations we establish a minimum principle of pontryagin s type to deal with the state constraints we introduce a penalty problem by using ekeland s principle the key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined conditions for normality of optimality conditions are given

on fuzzy rings abstract an internal description of the fuzzy subring fuzzy ideal generated by a finite fuzzy subset of a ring is provided also a fuzzy coset of a fuzzy ideal is defined and the set of all fuzzy cosets of a fuzzy ideal is given a ring structure a fuzzy semiprime ideal is also defined and some basic properties of this fuzzy ideal are studied

prime fuzzy ideals over noncommutative rings in this paper we introduce prime fuzzy ideals over a noncommutative ring this notion of primeness is equivalent to level cuts being crisp prime ideals it also generalizes the one provided by kumbhojkar and bapat in 1993 which lacks this equivalence in a noncommutative setting semiprime fuzzy ideals over a noncommutative ring are also defined and characterized as intersection of primes this allows us to introduce the fuzzy prime radical and contribute to establish the basis of a fuzzy noncommutative ring theory

prime l fuzzy ideals and primary l fuzzy ideals abstract in this paper the concepts of a primary l fuzzy ideal and a prime l fuzzy ideal and p primary are introduced and some fundamental propositions proved in particular a structural theorem for a primary l fuzzy ideal belonging to a prime l fuzzy ideal is given

optimal problems for nonlinear parabolic boundary control systems this paper considers optimal problems for boundary control systems with the control acting through a general nonlinear boundary condition the problems include constraints on the control and target conditions the final result is a version of pontryagin s maximum principle this setup covers such physical situations as stefan boltzmannboundary conditions and is based on an abstract theory of nonlinear programming problems