Fuzzy Prime Ideals Research Articles

(α,β)-fuzzy ideals of hemirings

We give a characterization of different types of ( α , β ) -fuzzy ideals of hemirings, where α , β ∈ { ∈ , q , ∈ ∨ q , ∈ ∧ q } and α ≠ ∈ ∧ q . Special attention is paid to ( ∈ , ∈ ∨ q ) -fuzzy prime and semiprime ideals.

Fuzzy radicals and prime fuzzy ideals of ordered semigroups

Let S be an ordered semigroup. A fuzzy subset of S is, by definition, an arbitrary mapping f: S → [0, 1], where [0, 1] is the usual interval of real numbers. Motivated by studying prime fuzzy ideals in rings, semigroups and ordered semigroups, and as a continuation of Kehayopulu and Tsingelis’s works in ordered semigroups in terms of fuzzy subsets, in this paper we introduce the notion of ordered fuzzy points of an ordered semigroup S, and give a characterization of prime fuzzy ideals of an ordered semigroup S. We also introduce the concepts of weakly prime fuzzy ideals, completely prime fuzzy ideals, completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of an ordered semigroup S, and establish the relationship between the five classes of ideals. Furthermore, we characterize weakly prime fuzzy ideals, completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of S by their level ideals. Finally, we introduce and investigate the fuzzy radicals of ordered semigroups by means of ordered fuzzy points, and prove that the fuzzy radical of every completely semiprime fuzzy ideal of an ordered semigroup S can be expressed as the intersection of all weakly completely prime fuzzy ideals containing it. As an application of the results of this paper, the corresponding results of semigroups (without order) are also obtained.

Fuzzy filters in ordered semigroups

In this paper we define fuzzy bi-ideal subsets and fuzzy bi-filters in ordered semigroups and characterize ordered semigroups in terms of fuzzy bi-ideal subsets and fuzzy bi-filters. We also define fuzzy left filters and fuzzy prime left ideals in ordered semigroups and characterize ordered semigroups in terms of these notions.

On fuzzy ideals of pseudo MV-algebras

Fuzzy ideals of pseudo MV-algebras are investigated. The homomorphic properties of fuzzy prime ideals are given. A one-to-one correspondence between the set of maximal ideals and the set of fuzzy maximal ideals satisfying (0) = 1 and (1) = 0 is obtained.

On the structure of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in commutative rings

This paper is a continuation of ideas presented by Davvaz [Roughness in rings, Inform. Sci., 164 (2004) 147–163; Roughness based on fuzzy ideals, Inform. Sci., 176 (2006) 2417–2437]. We introduce the notions of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in a ring, and give some properties of such ideals. Also, we discuss the relations between the upper and lower rough prime (primary) ideals and the upper and lower approximations of their homomorphism images.

The Zariski topology on the spectrum of prime L-submodules

Let R be a commutative ring with identity and let M be an R-module. We topologize L − Spec(M), the collection of all prime L-submodules of M, analogous to that for FSpec(R), the spectrum of fuzzy prime ideals of R, and investigate the properties of this topological space. In particular, we will study the relationship between L − Spec(M) and L − Spec(R/Ann(M)) and obtain some results.

Fuzzy prime ideals of pseudo-MV algebras

The aim of this paper is to introduce the notion of fuzzy prime ideals of pseudo-MV algebras and investigate some of its properties.

FUZZY PRIME IDEALS OF A SEMIRING AND FUZZY PRIME SUB-SEMIMODULES OF A SEMIMODULE OVER A SEMIRING

In this paper we define fuzzy prime right ideals and fuzzy prime subsemimodules of a right R-semimodule. We characterize those semirings for which each fuzzy ideal is prime and those semirings for which each fuzzy right ideal is prime.

FUZZY 2-(0-OR 1-)PRIME IDEALS IN SEMIRINGS

In this Paper three different types of fuzzy Prime ideals are introduced. Condition is obtained for a fuzzy 2-prime ideal will have two elements in its range. It has been shown that A is fuzzy 2-prime ideal of the semiring R if and only if 1-A is a fuzzy <TEX>$m_2-system$</TEX> in R.

Rough prime ideals and rough fuzzy prime ideals in semigroups

In this paper, we shall introduce the notions of rough prime ideals and rough fuzzy prime ideals in a semigroup, and give some properties of such ideals. Also, we will discuss the relations between the upper [lower] rough prime ideals and the upper [lower] approximations of their homomorphism images.

Cartesian Product on Fuzzy Prime Ideals

Cartesian Product on Fuzzy Prime Ideals

Fuzzy prime ideals and invertible fuzzy ideals in BCK-algebras

We state the relation between fuzzy prime ideals and invertible fuzzy ideals in commutative BCK-algebras.

PROPOSITIONAL LOGIC, FRAMES, AND FUZZY ALGEBRA

This paper offers a new look at such things as the fuzzy subalgebras and congruences of an algebra, the fuzzy ideals of a ring or a lattice, and similar entities, by exhibiting them as the models, in the chosen frame T of truth values, of naturally corresponding propositional theories. This provides a systematic approach to the study of the partially ordered sets formed by these various entities, and we demonstrate its usefulness by employing it to derive a number of results, some old and some new, concerning these partially ordered sets. In particular, we prove they are complete lattices, algebraic or continuous, depending on whether T is algebraic or continuous, respectively (Proposition 3); they satisfy the same lattice identities for arbitrary T that hold in the case T = 2 (Corollary of Proposition 4); and they are coherent frames for any coherent T whenever this is the case for T = 2 (Proposition 6). In addition we show, generalizing a result by Makamba and Murali [10], that the familiar classical situations where the congruences of an algebra correspond to certain other entities, such as the normal subgroups of a group or the ideals of a ring, extend to the fuzzy case by proving that the corresponding propositional theories are equivalent (Proposition 2). Further, we obtain the result of Gupta and Kantroo [5] that the fuzzy radical ideals of a commutative ring with unit are the meets of fuzzy prime ideals for arbitrary continuous T in place of the unit interval, using basic facts concerning continuous frames (Proposition 7).

Fractionary fuzzy ideals and Dedekind domains

We show that a finite-valued fractionary fuzzy ideal of R with invertible level ideals has a finite minimal generating set (as a fuzzy R-submodule). We also characterize Dedekind domains in terms of the factorization of fuzzy ideals as products of prime fuzzy ideals and also in terms of the invertibility of certain fractionary fuzzy ideals. We also examine the factorization of fractionary fuzzy ideals.

Fuzzy prime ideals and prime fuzzy ideals

In this paper, the concepts of fuzzy prime ideals and prime fuzzy ideals of a ring R are proposed, after analysing the deficiency of the fuzzy ideals defined in Malik and Mordeson (1990). Some characterizations and properties are discussed.

Some comments on spectrums of prime fuzzy ideals of a ring

Common and distinctive features of two prime spectrums, one obtained by Kumar and Kohli using 2-valued prime fuzzy ideals and the other obtained by the author using not-so-restrictive definition of prime fuzzy ideals, are discussed. It is also shown that some results claimed by Kumar and Kohli, regarding spectrum of 2-valued prime fuzzy ideals, need some corrections.

Fuzzy implicative and Boolean ideals of MV-algebras

Abstract We show how to obtain results in MV -algebras by considering their fuzzy ideals. In particular we prove the following theorem: Let μ be a fuzzy ideal of an MV -algebra X . Then the folllowing are equivalent: (1) μ (x ∧ x = μ(0) for all x ϵ X , (2) μ ( x n ) = μ ( x ) for all x ϵ X and all n ⩾ 1, (3) X / X μ is a Boolean algebra, (4) μ is fuzzy implicative. If I is an ideal of X and we take μ = χ I then we obtain corresponding results for the ideal I and X . We show how to extend fuzzy prime and fuzzy implicative ideals. Various fuzzy radicals are considered and related to the corresponding radicals of the algebra, and we show how they live in the spectrum of the algebra.

Spectrum of prime fuzzy ideals

A topological space called a spectrum of prime fuzzy ideals and denoted by Fspec( R) is obtained by introducing a topology on the set of prime fuzzy ideals of a ring R. The prime fuzzy ideals used here (introduced by the author elsewhere), are not necessarily two valued, the topology introduced is compact and the usual (nonfuzzy) prime spectrum Spec( R) is a subspace of Fspec( R) and dense in it. Situations under which fuzzy spectrums of different rings are homeomorphic are discussed. In particular, Fspec( R) is proved to be homeomorphic to Fspec( R/Frad( R)), where Frad( R) is the fuzzy prime radical of R. The correspondence associating a ring R to the topological space Fspec( R) is shown to define a contravarient functor from the category of rings with unity, into the category of compact topological spaces.

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.

Characterizations of monoids by the properties of their fuzzy subsystems

In this paper we initiate the study of fuzzy subsystems of an S -system, where S is a monoid with zero and S -systems are representations of S . We introduce the notions of pure fuzzy, purely fuzzy maximal and purely fuzzy prime ideals of a monoid S . It is shown that the set of purely fuzzy prime ideals of S admits the structure of a topological space, called the pure fuzzy spectrum of S . We also define a pure fuzzy subsystem of an S -system and call an S -system fuzzy normal if each of its subsystems is pure fuzzy. Monoids of which all S -systems are fuzzy normal are characterized. It is shown, among other results, that such monoids are right weakly regular.