scholarly journals Fuzzy Prime Ideal Theorem in Residuated Lattices

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Pierre Carole Kengne ◽  
Blaise Blériot Koguep ◽  
Celestin Lele
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Prime Ideals ◽  
Fuzzy Subset ◽  
Different Types ◽  
Fuzzy Ideal ◽  
Prime Ideal Theorem

This paper mainly focuses on building the fuzzy prime ideal theorem of residuated lattices. Firstly, we introduce the notion of fuzzy ideal generated by a fuzzy subset of a residuated lattice and we give a characterization. Also, we introduce different types of fuzzy prime ideals and establish existing relationships between them. We prove that any fuzzy maximal ideal is a fuzzy prime ideal in residuated lattice. Finally, we give and prove the fuzzy prime ideal theorem in residuated lattice.

Ideals of Residuated Lattices

2021 ◽  
Vol 58 (2) ◽  
pp. 182-205
Author(s):  
Liviu-Constantin Holdon ◽  
Arsham Borumand Saeid
Keyword(s):  
Prime Ideal ◽  
Residuated Lattice ◽  
Extension Property ◽  
Characterization Theorem ◽  
Residuated Lattices ◽  
Quotient Algebra ◽  
Prime Ideals ◽  
Characterization Theorems

In this article, we study ideals in residuated lattice and present a characterization theorem for them. We investigate some related results between the obstinate ideals and other types of ideals of a residuated lattice, likeness Boolean, primary, prime, implicative, maximal and ʘ-prime ideals. Characterization theorems and extension property for obstinate ideal are stated and proved. For the class of ʘ-residuated lattices, by using the ʘ-prime ideals we propose a characterization, and prove that an ideal is an ʘ-prime ideal iff its quotient algebra is an ʘ-residuated lattice. Finally, by using ideals, the class of Noetherian (Artinian) residuated lattices is introduced and Cohen’s theorem is proved.

Minimal prime ideals and arithmetic comprehension

1991 ◽  
Vol 56 (1) ◽  
pp. 67-70 ◽  
Author(s):  
Kostas Hatzikiriakou
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Commutative Rings ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Order Arithmetic ◽  
Minimal Prime ◽  
Weak König’S Lemma ◽  
König’S Lemma

We assume that the reader is familiar with the program of “reverse mathematics” and the development of countable algebra in subsystems of second order arithmetic. The subsystems we are using in this paper are RCA0, WKL0 and ACA0. (The reader who wants to learn about them should study [1].) In [1] it was shown that the statement “Every countable commutative ring has a prime ideal” is equivalent to Weak Konig's Lemma over RCA0, while the statement “Every countable commutative ring has a maximal ideal” is equivalent to Arithmetic Comprehension over RCA0. Our main result in this paper is that the statement “Every countable commutative ring has a minimal prime ideal” is equivalent to Arithmetic Comprehension over RCA0. Minimal prime ideals play an important role in the study of countable commutative rings; see [2, pp. 1–7].

scholarly journals On 2-absorbing ideals of commutative rings

2007 ◽  
Vol 75 (3) ◽  
pp. 417-429 ◽  
Author(s):  
Ayman Badawi
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Integral Domain ◽  
Commutative Rings ◽  
Dedekind Domain ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Maximal Ideals ◽  
Noetherian Domain ◽  
Valuation Domains

Suppose that R is a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of 2-absorbing ideal which is a generalisation of prime ideal. A nonzero proper ideal I of R is called a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. It is shown that a nonzero proper ideal I of R is a 2-absorbing ideal if and only if whenever I1I2I3 ⊆ I for some ideals I1,I2,I3 of R, then I1I2 ⊆ I or I2I3 ⊆ I or I1I3 ⊆ I. It is shown that if I is a 2-absorbing ideal of R, then either Rad(I) is a prime ideal of R or Rad(I) = P1 ⋂ P2 where P1,P2 are the only distinct prime ideals of R that are minimal over I. Rings with the property that every nonzero proper ideal is a 2-absorbing ideal are characterised. All 2-absorbing ideals of valuation domains and Prüfer domains are completely described. It is shown that a Noetherian domain R is a Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R. If RM is Noetherian for each maximal ideal M of R, then it is shown that an integral domain R is an almost Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R.

scholarly journals IDEAL PRIMA FUZZY DI SEMIGRUP S

2017 ◽  
Vol 2 (2) ◽  
pp. 46
Author(s):  
Abdurahim Abdurahim
Keyword(s):  
Prime Ideal ◽  
Membership Function ◽  
Closed Interval ◽  
Fuzzy Membership ◽  
Fuzzy Membership Function ◽  
Fuzzy Subset ◽  
Fuzzy Ideal ◽  
Function Domain

A fuzzy membership function is a function that maps the nonempty set  to a closed interval . Furthermore, if the function domain is replaced with a semigroup, then the function is called fuzzy subset. A fuzzy subset mapping  to  is denoted by . A fuzzy subset  is called fuzzy ideal if it satisfies both  and . Moreover,  is called a fuzzy prime ideal if for any fuzzy ideal  and , with  implies  or . In this paper be investigated about some characteristics of prime fuzzy ideals and some example of them.

scholarly journals On 2-absorbing ideals of commutative semirings

2019 ◽  
Vol 19 (02) ◽  
pp. 2050034
Author(s):  
H. Behzadipour ◽  
P. Nasehpour
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Ideal Formula ◽  
Minimal Prime ◽  
Unique Maximal Ideal

In this paper, we investigate 2-absorbing ideals of commutative semirings and prove that if [Formula: see text] is a nonzero proper ideal of a subtractive valuation semiring [Formula: see text] then [Formula: see text] is a 2-absorbing ideal of [Formula: see text] if and only if [Formula: see text] or [Formula: see text] where [Formula: see text] is a prime ideal of [Formula: see text]. We also show that each 2-absorbing ideal of a subtractive semiring [Formula: see text] is prime if and only if the prime ideals of [Formula: see text] are comparable and if [Formula: see text] is a minimal prime over a 2-absorbing ideal [Formula: see text], then [Formula: see text], where [Formula: see text] is the unique maximal ideal of [Formula: see text].

scholarly journals On Fuzzy Ideals ofBL-Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Biao Long Meng ◽  
Xiao Long Xin
Keyword(s):  
Prime Ideal ◽  
Distributive Lattice ◽  
Fuzzy Set ◽  
Representation Theorem ◽  
Prime Ideals ◽  
Converse Implication ◽  
Fuzzy Ideal ◽  
Stone Representation

In this paper we investigate further properties of fuzzy ideals of aBL-algebra. The notions of fuzzy prime ideals, fuzzy irreducible ideals, and fuzzy Gödel ideals of aBL-algebra are introduced and their several properties are investigated. We give a procedure to generate a fuzzy ideal by a fuzzy set. We prove that every fuzzy irreducible ideal is a fuzzy prime ideal but a fuzzy prime ideal may not be a fuzzy irreducible ideal and prove that a fuzzy prime idealωis a fuzzy irreducible ideal if and only ifω0=1and|Im⁡(ω)|=2. We give the Krull-Stone representation theorem of fuzzy ideals inBL-algebras. Furthermore, we prove that the lattice of all fuzzy ideals of aBL-algebra is a complete distributive lattice. Finally, it is proved that every fuzzy Boolean ideal is a fuzzy Gödel ideal, but the converse implication is not true.

scholarly journals Characterizations of Fuzzy Ideals in Coresiduated Lattices

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Yan Liu ◽  
Mucong Zheng
Keyword(s):  
Prime Ideal ◽  
Prime Ideals ◽  
Fuzzy Ideal

The notions of fuzzy ideals are introduced in coresiduated lattices. The characterizations of fuzzy ideals, fuzzy prime ideals, and fuzzy strong prime ideals in coresiduated lattices are investigated and the relations between ideals and fuzzy ideals are established. Moreover, the equivalence of fuzzy prime ideals and fuzzy strong prime ideals is proved in prelinear coresiduated lattices. Furthermore, the conditions under which a fuzzy prime ideal is derived from a fuzzy ideal are presented in prelinear coresiduated lattices.

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 Proper Fuzzification of Prime Ideals of a Hemiring

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
H. V. Kumbhojkar
Keyword(s):  
Prime Ideals ◽  
Different Types ◽  
Fuzzy Ideal ◽  
Nil Radical

Prime fuzzy ideals, prime fuzzyk-ideals, and prime fuzzyh-ideals are roped in one condition. It is shown that this way better fuzzification is achieved. Other major results of the paper are: every fuzzy ideal (resp.,k-ideal,h-ideal) is contained in a prime fuzzy ideal (resp.,k-ideal,h-ideal). Prime radicals and nil radicals of a fuzzy ideal are defined; their relationship is established. The nil radical of a fuzzyk-ideal (resp., anh-ideal) is proved to be a fuzzyk-ideal (resp.,h-ideal). The correspondence theorems for different types of fuzzy ideals of hemirings are established. The concept of primary fuzzy ideal is introduced. Minimum imperative for proper fuzzification is suggested and it is shown that the fuzzifications introduced in this paper are proper fuzzifications.

scholarly journals On the essentiality and primeness of λ-super socle of C(X)

2018 ◽  
Vol 19 (2) ◽  
pp. 261
Author(s):  
S. Mehran ◽  
M. Namdari ◽  
S. Soltanpour
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Topological Property ◽  
Regular Cardinal ◽  
Cardinal Number ◽  
Prime Ideals

<p>Spaces X for which the annihilator of Sλ(X), the λ-super socle of C(X) (i.e., the set of elements of C(X) that cardinality of their cozerosets are less than λ, where λ is a regular cardinal number such that λ≤|X|) is generated by an idempotent are characterized. This enables us to find a topological property equivalent to essentiality of Sλ(X). It is proved that every prime ideal in C(X) containing Sλ(X) is essential and it is an intersection of free prime ideals. Primeness of Sλ(X) is characterized via a fixed maximal ideal of C(X).</p>

Sign in / Sign up

Export Citation Format

Share Document