scholarly journals A Certain Structure of Bipolar Fuzzy Subrings

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1397
Author(s):  
Hanan Alolaiyan ◽  
Muhammad Haris Mateen ◽  
Dragan Pamucar ◽  
Muhammad Khalid Mahmmod ◽  
Farrukh Arslan
Keyword(s):  
Fuzzy Set ◽  
Quotient Ring ◽  
Ring Homomorphism ◽  
Universal Relation ◽  
Support Set ◽  
Ring Isomorphism ◽  
Quotient Rings ◽  
Fundamental Theorems ◽  
Fuzzy Ring

The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define (α,β)-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism.

scholarly journals On Fundamental Theorems of Fuzzy Isomorphism of Fuzzy Subrings over a Certain Algebraic Product

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 998
Author(s):  
Alaa Altassan ◽  
Muhammad Haris Mateen ◽  
Dragan Pamucar
Keyword(s):  
Fuzzy Set ◽  
Significant Relationship ◽  
Quotient Ring ◽  
Surjective Homomorphism ◽  
Support Set ◽  
Fuzzy Ideal ◽  
Quotient Rings ◽  
Fundamental Theorems ◽  
The Given

In this study, we define the concept of an ω-fuzzy set ω-fuzzy subring and show that the intersection of two ω-fuzzy subrings is also an ω-fuzzy subring of a given ring. Moreover, we give the notion of an ω-fuzzy ideal and investigate different fundamental results of this phenomenon. We extend this ideology to propose the notion of an ω-fuzzy coset and develop a quotient ring with respect to this particular fuzzy ideal analog into a classical quotient ring. Additionally, we found an ω-fuzzy quotient subring. We also define the idea of a support set of an ω-fuzzy set and prove various important characteristics of this phenomenon. Further, we describe ω-fuzzy homomorphism and ω-fuzzy isomorphism. We establish an ω-fuzzy homomorphism between an ω-fuzzy subring of the quotient ring and an ω-fuzzy subring of this ring. We constitute a significant relationship between two ω-fuzzy subrings of quotient rings under the given ω-fuzzy surjective homomorphism and prove some more fundamental theorems of ω-fuzzy homomorphism for these specific fuzzy subrings. Finally, we present three fundamental theorems of ω-fuzzy isomorphism.

scholarly journals FUZZY RINGS AND ITS PROPERTIES

2017 ◽  
Vol 5 (1) ◽  
pp. 32
Author(s):  
Karyati Karyati ◽  
Rifki Chandra Utama
Keyword(s):  
Normal Subgroup ◽  
Fuzzy Sets ◽  
Algebraic Structure ◽  
Binary Operation ◽  
Quotient Ring ◽  
Ring Homomorphism ◽  
Factor Group ◽  
Semi Group ◽  
Fuzzy Ideal ◽  
Fuzzy Ring

Abstract One of algebraic structure that involves a binary operation is a group that is defined  an un empty set (classical) with an associative binary operation, it has identity elements and each element has an inverse. In the structure of the group known as the term subgroup, normal subgroup, subgroup and factor group homomorphism and its properties. Classical algebraic structure is developed to algebraic structure fuzzy by the researchers as an example semi group fuzzy and fuzzy group after fuzzy sets is introduced by L. A. Zadeh at 1965. It is inspired of writing about semi group fuzzy and group of fuzzy, a research on the algebraic structure of the ring is held with reviewing ring fuzzy, ideal ring fuzzy, homomorphism ring fuzzy and quotient ring fuzzy with its properties. The results of this study are obtained fuzzy properties of the ring, ring ideal properties fuzzy, properties of fuzzy ring homomorphism and properties of fuzzy quotient ring by utilizing a subset of a subset level  and strong level  as well as image and pre-image homomorphism fuzzy ring. Keywords: fuzzy ring, subset level, homomorphism fuzzy ring, fuzzy quotient ring

scholarly journals It takes two to tango: the interplay between decision logics, communication strategies and social media engagement in start-ups

2021 ◽  
Author(s):  
Christian Rudeloff ◽  
Stefanie Pakura ◽  
Fabian Eggers ◽  
Thomas Niemand
Keyword(s):  
Social Media ◽  
Comparative Analysis ◽  
Fuzzy Set ◽  
Exploratory Study ◽  
Communication Strategies ◽  
Qualitative Comparative Analysis ◽  
Communication Models ◽  
Social Media Engagement ◽  
Start Ups

AbstractThis manuscript analyzes start-ups’ usage of different communication strategies (information, response, involvement), their underlying decision logics (effectuation, causation, strategy absence) and respective social media success. A multitude of studies have been published on the decision logics of entrepreneurs as well as on different communication strategies. Decision logics and according strategies and actions are closely connected. Still, research on the interplay between the two areas is largely missing. This applies in particular to the effect of different decision logics and communication models on social media success. Through a combination of case studies with fuzzy-set Qualitative Comparative Analysis this exploratory study demonstrates that different combinations of causal and absence of strategy decision logics can be equally successful when it comes to social media engagement, whereas effectuation is detrimental for success. Furthermore, we find that two-way-communication is essential to create engagement, while information strategy alone cannot lead to social media success. This study provides new insights into the role of decision logics and connects effectuation theory with the communication literature, a field that has been dominated by causal approaches.

A Generalization of the Concept of a Ring of Quotients

1971 ◽  
Vol 14 (4) ◽  
pp. 517-529 ◽  
Author(s):  
John K. Luedeman
Keyword(s):  
Left Ideal ◽  
Quotient Ring ◽  
Original Method ◽  
Duke Math ◽  
Injective Hull ◽  
Direct Products ◽  
Left Module ◽  
Matrix Rings ◽  
Ring Of Quotients ◽  
Quotient Rings

AbstractSanderson (Canad. Math. Bull., 8 (1965), 505–513), considering a nonempty collection Σ of left ideals of a ring R, with unity, defined the concepts of “Σ-injective module” and “Σ-essential extension” for unital left modules. Letting Σ be an idempotent topologizing set (called a σ-set below) Σanderson proved the existence of a “Σ-injective hull” for any unital left module and constructed an Utumi Σ-quotient ring of R as the bicommutant of the Σ-injective hull of RR. In this paper, we extend the concepts of “Σinjective module”, “Σ-essentialextension”, and “Σ-injective hull” to modules over arbitrary rings. An overring Σ of a ring R is a Johnson (Utumi) left Σ-quotient ring of R if RR is Σ-essential (Σ-dense) in RS. The maximal Johnson and Utumi Σ-quotient rings of R are constructed similar to the original method of Johnson, and conditions are given to insure their equality. The maximal Utumi Σquotient ring U of R is shown to be the bicommutant of the Σ-injective hull of RR when R has unity. We also obtain a σ-set UΣ of left ideals of U, generated by Σ, and prove that Uis its own maximal Utumi UΣ-quotient ring. A Σ-singular left ideal ZΣ(R) of R is defined and U is shown to be UΣ-injective when Z Σ(R) = 0. The maximal Utumi Σ-quotient rings of matrix rings and direct products of rings are discussed, and the quotient rings of this paper are compared with these of Gabriel (Bull. Soc. Math. France, 90 (1962), 323–448) and Mewborn (Duke Math. J. 35 (1968), 575–580). Our results reduce to those of Johnson and Utumi when 1 ∊ R and Σ is taken to be the set of all left ideals of R.

Semicritical Rings and the Quotient Problem

1984 ◽  
Vol 27 (2) ◽  
pp. 160-170
Author(s):  
Karl A. Kosler
Keyword(s):  
Quotient Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Regular Elements ◽  
The Right ◽  
Minimal Prime ◽  
Necessary And Sufficient ◽  
Quotient Rings ◽  
The Relationship

AbstractThe purpose of this paper is to examine the relationship between the quotient problem for right noetherian nonsingular rings and the quotient problem for semicritical rings. It is shown that a right noetherian nonsingular ring R has an artinian classical quotient ring iff certain semicritical factor rings R/Ki, i = 1,…,n, possess artinian classical quotient rings and regular elements in R/Ki lift to regular elements of R for all i. If R is a two sided noetherian nonsingular ring, then the existence of an artinian classical quotient ring is equivalent to each R/Ki possessing an artinian classical quotient ring and the right Krull primes of R consisting of minimal prime ideals. If R is also weakly right ideal invariant, then the former condition is redundant. Necessary and sufficient conditions are found for a nonsingular semicritical ring to have an artinian classical quotient ring.

Quotient Rings of a Class of Lattice-Ordered-Rings

1973 ◽  
Vol 25 (3) ◽  
pp. 627-645 ◽  
Author(s):  
Stuart A. Steinberg
Keyword(s):  
Identity Element ◽  
Quotient Ring ◽  
Ring Extension ◽  
Qf Ring ◽  
Left Quotient ◽  
Quotient Rings

An f-ring R with zero right annihilator is called a qf-ring if its Utumi maximal left quotient ring Q = Q(R) can be made into and f-ring extension of R. F. W. Anderson [2, Theorem 3.1] has characterized unital qf-rings with the following conditions: For each q ∈ Q and for each pair d1, d2 ∈ R+ such that diq ∈ R(i) (d1q)+ Λ (d2q)- = 0, and(ii) d1 Λ d2 = 0 implies (d1q)+ Λ d2 = 0.We remark that this characterization holds even when R does not have an identity element.

scholarly journals Maximal quotient rings of prime group algebras. II Uniform right ideals

1977 ◽  
Vol 24 (3) ◽  
pp. 339-349 ◽  
Author(s):  
John Hannah
Keyword(s):  
Group Algebra ◽  
Quotient Ring ◽  
Group Algebras ◽  
Normal Subgroups ◽  
Locally Finite ◽  
Residually Finite ◽  
Zero Characteristic ◽  
Quotient Rings

AbstractSuppose KG is a prime nonsingular group algebra with uniform right ideals. We show that G has no nontrivial locally finite normal subgroups. If G is soluble or residually finite, or if K has zero characteristic and G is linear, then the maximal right quotient ring of KG is simple Artinian.

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