Fuzzy Ideals Of Ring Research Articles

Applications of Fuzzy Semiprimary Ideals under Group Action

Group actions are a valuable tool for investigating the symmetry and automorphism features of rings. The concept of fuzzy ideals in rings has been expanded with the introduction of fuzzy primary, weak primary, and semiprimary ideals. This paper explores the existence of fuzzy ideals that are semiprimary but neither weak primary nor primary. Furthermore, it defines a group action on a fuzzy ideal and examines the properties of fuzzy ideals and their level cuts under this group action. In fact, it aims to investigate the relationship between fuzzy semiprimary ideals and the radical of fuzzy ideals under group action. Additionally, it includes the results related to the radical of fuzzy ideals and fuzzy G-semiprimary ideals. Moreover, the preservation of the image and inverse image of a fuzzy G-semiprimary ideal of a ring R under certain conditions is also studied. It delves into the algebraic nature of fuzzy ideals and the radical under G-homomorphism of fuzzy ideals.

English

English

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 SEMI-IDEAL AND GENERALIZED FUZZY QUOTIENT RING

The concepts of fuzzy semi-ideals of R with respect to HR ≤ and generalized fuzzy quotient rings are introduced. Some properties of fuzzy semi- ideals are discussed. Finally, several isomorphism theorems for generalized fuzzy quotient rings are established. 1. Preliminaries Since the concept of fuzzy subgroups was introduced by Rosenfeld (6) in 1971, the literature of these fuzzy algebraic concepts has been growing very rapidly. Liu (4) defined and studied fuzzy subrings as well as fuzzy ideals in rings. Subsequently, Mukherjee and Sen (5), and Zhang (7) further various fuzzy characterized ideals in rings. Based on the concept of fuzzy ideal, Kumbhojkar and Bapat (3) introduced the concept of fuzzy quotient ring and established the correspondence theorems for fuzzy ideals. In this paper, we first generalize the concepts of fuzzy ideals and fuzzy quotient rings and then establish several isomorphism theorems for generalized fuzzy quotient rings.

Intuitionistic nil radicals of intuitionistic fuzzy ideals and Euclidean intuitionistic fuzzy ideals in rings

The notion of intuitionistic nil radicals of intuitionistic fuzzy ideals in rings is introduced, and related properties are investigated. The notion of semiprime intuitionistic fuzzy ideals is provided, and its characterization is established. The concept of Euclidean intuitionistic fuzzy ideals is also introduced, and its characterization is established.

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).

Primary fuzzy ideals and radical of fuzzy ideals

Abstract The aim of this paper is to introduce primary fuzzy ideals and radical of fuzzy ideals in rings.

Prime fuzzy ideals in rings

In this paper give a characterization of prime fuzzy ideals. We introduce weakly completely prime fuzzy ideals and establish a relation between completely prime fuzzy ideals and weakly completely prime fuzzy ideals.