scholarly journals Fuzzy Normed Rings

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 515 ◽  
Author(s):  
Aykut Emniyet ◽  
Memet Şahin
Keyword(s):  
Prime Ideal ◽  
Fuzzy Sets ◽  
Maximal Ideal ◽  
Ring Homomorphism ◽  
Ring Theory ◽  
Basic Properties ◽  
Normed Ring ◽  
Normed Ideal ◽  
Algebraic Properties ◽  
Definition Of

In this paper, the concept of fuzzy normed ring is introduced and some basic properties related to it are established. Our definition of normed rings on fuzzy sets leads to a new structure, which we call a fuzzy normed ring. We define fuzzy normed ring homomorphism, fuzzy normed subring, fuzzy normed ideal, fuzzy normed prime ideal, and fuzzy normed maximal ideal of a normed ring, respectively. We show some algebraic properties of normed ring theory on fuzzy sets, prove theorems, and give relevant examples.

scholarly journals Intuitionistic Fuzzy Normed Subrings and Intuitionistic Fuzzy Normed Ideals

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1594
Author(s):  
Nour Abed Alhaleem ◽  
Abd Ghafur Ahmad
Keyword(s):  
Inverse Image ◽  
Ring Homomorphism ◽  
Basic Properties ◽  
Intuitionistic Fuzzy ◽  
Normed Ring

The main goal of this paper is to introduce the notion of intuitionistic fuzzy normed rings and to establish basic properties related to it. We extend normed rings by incorporating the idea of intuitionistic fuzzy to normed rings, we develop a new structure of fuzzy rings which will be called an intuitionistic fuzzy normed ring. As an extension of intuitionistic fuzzy normed rings, we define the concept of intuitionistic fuzzy normed subrings and intuitionistic fuzzy normed ideals. Some essential operations specially subset, complement, union, intersection and several properties relating to the notion of generalized intuitionistic fuzzy normed rings are identified. Homomorphism and isomorphism of intuitionistic fuzzy normed subrings are characterized. We identify the image and the inverse image of intuitionistic fuzzy normed subrings under ring homomorphism and study their elementary properties. Some properties of intuitionistic fuzzy normed rings and relevant examples are presented.

On EMV-Semirings

2019 ◽  
Vol 69 (4) ◽  
pp. 739-752 ◽  
Author(s):  
R. A. Borzooei ◽  
M. Shenavaei ◽  
A. Di Nola ◽  
O. Zahiri
Keyword(s):  
Distributive Lattice ◽  
Algebraic Structure ◽  
Maximal Ideal ◽  
Boolean Algebras ◽  
Algebraic Extension ◽  
Theoretic Approach ◽  
Basic Properties ◽  
Mv Algebra ◽  
Definition Of ◽  
Generalized Boolean Algebras

Abstract The paper deals with an algebraic extension of MV-semirings based on the definition of generalized Boolean algebras. We propose a semiring-theoretic approach to EMV-algebras based on the connections between such algebras and idempotent semirings. We introduce a new algebraic structure, not necessarily with a top element, which is called an EMV-semiring and we get some examples and basic properties of EMV-semiring. We show that every EMV-semiring is an EMV-algebra and every EMV-semiring contains an MV-semiring and an MV-algebra. Then, we study EMV-semiring as a lattice and prove that any EMV-semiring is a distributive lattice. Moreover, we define an EMV-semiring homomorphism and show that the categories of EMV-semirings and the category of EMV-algebras are isomorphic. We also define the concepts of GI-simple and DLO-semiring and prove that every EMV-semiring is a GI-simple and a DLO-semiring. Finally, we propose a representation for EMV-semirings, which proves that any EMV-semiring is either an MV-semiring or can be embedded into an MV-semiring as a maximal ideal.

scholarly journals Approximaitly Prime Submodules and Some Related Concepts

2019 ◽  
Vol 32 (2) ◽  
pp. 103
Author(s):  
Ali Sh. Ajeel ◽  
Haibat K. Mohammad Ali
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Research Note ◽  
Prime Radical ◽  
Basic Properties ◽  
Prime Submodules ◽  
Prime Submodule ◽  
Definition Of

In this research note approximately prime submodules is defined as a new generalization of prime submodules of unitary modules over a commutative ring with identity. A proper submodule  of an -module  is called an approximaitly prime submodule of  (for short app-prime submodule), if when ever , where , , implies that either  or . So, an ideal  of a ring  is called app-prime ideal of  if   is an app-prime submodule of -module . Several basic properties, characterizations and examples of approximaitly prime submodules were given. Furthermore, the definition of approximaitly prime radical of submodules of modules were introduced, and some of it is properties were established.

scholarly journals One sided invertibility and localisation

1992 ◽  
Vol 34 (3) ◽  
pp. 333-339 ◽  
Author(s):  
C. R. Hajarnavis
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Polynomial Identity ◽  
Noetherian Ring ◽  
Crucial Role ◽  
Prime Ideals ◽  
Careful Study ◽  
Definition Of ◽  
The Right

In general, a prime ideal P of a prime Noetherian ring need not be classically localisable. Since such a localisation, when it does exist, is a striking property; sufficiency criteria which guarantee it are worthy of careful study. One such condition which ensures localisation is when P is an invertible ideal [5, Theorem 1.3]. The known proofs of this result utilise both the left as well as the right invertiblity of P. Such a requirement is, in practice, somewhat restrictive. There are many occasions such as when a product of prime ideals is invertible [6] or when a non-idempotent maximal ideal is known to be projective only on one side [2], when the assumptions lead to invertibilty also on just one side. Our main purpose here is to show that in the context of Noetherian prime polynomial identity rings, this one-sided assumption is enough to ensure classical localisation [Theorem 3.5]. Consequently, if a maximal ideal in such a ring is invertible on one side then it is invertible on both sides [Proposition 4.1]. This result plays a crucial role in [2]. As a further application we show that for polynomial identity rings the definition of a unique factorisation ring is left-right symmetric [Theorem 4.4].

scholarly journals Intuitionistic fuzzy normed prime and maximal ideals

2021 ◽  
Vol 6 (10) ◽  
pp. 10565-10580
Author(s):  
Nour Abed Alhaleem ◽  
◽  
Abd Ghafur Ahmad
Keyword(s):  
Characteristic Function ◽  
Prime Ideal ◽  
Maximal Ideal ◽  
Prime Ideals ◽  
Intuitionistic Fuzzy ◽  
Normed Ideal ◽  
Maximal Ideals

<abstract><p>Motivated by the new notion of intuitionistic fuzzy normed ideal, we present and investigate some associated properties of intuitionistic fuzzy normed ideals. We describe the intrinsic product of any two intuitionistic fuzzy normed subsets and show that the intrinsic product of intuitionistic fuzzy normed ideals is a subset of the intersection of these ideals. We specify the notions of intuitionistic fuzzy normed prime ideal and intuitionistic fuzzy normed maximal ideal, we present the conditions under which a given intuitionistic fuzzy normed ideal is considered to be an intuitionistic fuzzy normed prime (maximal) ideal. In addition, the relation between the intuitionistic characteristic function and prime and maximal ideals is generalized. Finally, we characterize relevant properties of intuitionistic fuzzy normed prime ideals and intuitionistic fuzzy normed maximal ideals.</p></abstract>

scholarly journals On the Most Extended Modal Operator of First Type over Interval-Valued Intuitionistic Fuzzy Sets

Mathematics ◽  
2018 ◽  
Vol 6 (7) ◽  
pp. 123 ◽  
Author(s):  
Krassimir Atanassov
Keyword(s):  
Fuzzy Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Modal Operator ◽  
Basic Properties ◽  
Intuitionistic Fuzzy ◽  
Definition Of ◽  
Interval Valued

The definition of the most extended modal operator of first type over interval-valued intuitionistic fuzzy sets is given, and some of its basic properties are studied.

Sifat-sifat Submodul Prima dan Submodul Prima Lemah

2020 ◽  
Vol 2 (2) ◽  
pp. 183
Author(s):  
Hisyam Ihsan ◽  
Muhammad Abdy ◽  
Samsu Alam B
Keyword(s):  
Prime Ideal ◽  
Literature Study ◽  
Prime Submodules ◽  
Unitary Module ◽  
Prime Submodule ◽  
Definition Of ◽  
The Relationship

Penelitian ini merupakan penelitian kajian pustaka yang bertujuan untuk mengkaji sifat-sifat submodul prima dan submodul prima lemah serta hubungan antara keduanya. Kajian dimulai dari definisi submodul prima dan submodul prima lemah, selanjutnya dikaji mengenai sifat-sifat dari keduanya. Pada penelitian ini, semua ring yang diberikan adalah ring komutatif dengan unsur kesatuan dan modul yang diberikan adalah modul uniter. Sebagai hasil dari penelitian ini diperoleh beberapa pernyataan yang ekuivalen, misalkan  suatu -modul ,  submodul sejati di  dan ideal di , maka ketiga pernyataan berikut ekuivalen, (1)  merupakan submodul prima, (2) Setiap submodul tak nol dari   -modul memiliki annihilator yang sama, (3) Untuk setiap submodul  di , subring  di , jika berlaku  maka  atau . Di lain hal, pada submodul prima lemah jika diberikan  suatu -modul,  submodul sejati di , maka pernyataan berikut ekuivalen, yaitu (1) Submodul  merupakan submodul prima lemah, (2) Untuk setiap , jika  maka . Selain itu, didapatkan pula hubungan antara keduanya, yaitu setiap submodul prima merupakan submodul prima lemah.Kata Kunci: Submodul Prima, Submodul Prima Lemah, Ideal Prima. This research is literature study that aims to examine the properties of prime submodules and weakly prime submodules and the relationship between  both of them. The study starts from the definition of prime submodules and weakly prime submodules, then reviewed about the properties both of them. Throughout this paper all rings are commutative with identity and all modules are unitary. As the result of this research, obtained several equivalent statements, let  be a -module,  be a proper submodule of  and  ideal of , then the following three statetments are equivalent, (1)  is a prime submodule, (2) Every nonzero submodule of   -module has the same annihilator, (3) For any submodule  of , subring  of , if  then  or . In other case, for weakly prime submodules, if given  is a unitary -module,  be a proper submodule of , then the following statements are equivalent, (1)  is a weakly prime submodule, (2) For any , if  then . In addition, also found the relationship between both of them, i.e. any prime submodule is weakly prime submodule.Keywords: Prime Submodules, Weakly Prime Submdules, Prime Ideal.

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

scholarly journals L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras

2019 ◽  
Vol 17 (1) ◽  
pp. 1538-1546
Author(s):  
Xin Zhou ◽  
Liangyun Chen ◽  
Yuan Chang
Keyword(s):  
Fuzzy Sets ◽  
Homomorphic Image ◽  
Quotient Algebra ◽  
Algebraic Properties ◽  
Novikov Algebras ◽  
Fuzzy Ideal

Abstract In this paper, we apply the concept of fuzzy sets to Novikov algebras, and introduce the concepts of L-fuzzy ideals and L-fuzzy subalgebras. We get a sufficient and neccessary condition such that an L-fuzzy subspace is an L-fuzzy ideal. Moreover, we show that the quotient algebra A/μ of the L-fuzzy ideal μ is isomorphic to the algebra A/Aμ of the non-fuzzy ideal Aμ. Finally, we discuss the algebraic properties of surjective homomorphic image and preimage of an L-fuzzy ideal.

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