zDECSAI, Universidad de Granadagnavarro/files/fuzzy_slides.pdf · 2015-09-09 · Prime fuzzy ideals...
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Prime fuzzy ideals over noncommutative rings
F. J. Lobillo†, O. Cortadellas† and G. Navarro‡
†Departamento de Álgebra, Universidad de Granada
‡DECSAI, Universidad de Granada
XI Jornadas de Teoría de Anillos, June 2nd, 2012
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Work published in:Fuzzy Sets and Systems, 199 (2012) 108�120
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Index
1 Introduction and History
2 A survey on fuzzy prime ideals
3 Fuzzy primeness over noncommutative rings
4 Fuzzy semiprimeness and fuzzy prime radical
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Fuzzy sets (main aim)
Mathematical treatment of uncertainty, ambiguity, subjectivity,...
Measuring the warm feeling
Is the water warm?
T < 10◦ Sure not!
10◦ < T < 20◦ it could be...
20◦ < T < 30◦ it seems so...
30◦ < T < 40◦ it could be...
40◦ < T Too hot!
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Fuzzy sets (main aim)
Mathematical treatment of uncertainty, ambiguity, subjectivity,...
Measuring the warm feeling
Is the water warm?
T < 10◦ Sure not!
10◦ < T < 20◦ it could be...
20◦ < T < 30◦ it seems so...
30◦ < T < 40◦ it could be...
40◦ < T Too hot!
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Fuzzy sets (main aim)
Mathematical treatment of uncertainty, ambiguity, subjectivity,...
Measuring the warm feeling
Is the water warm?
T < 10◦ → 0% warm
10◦ < T < 20◦ → 50% warm
20◦ < T < 30◦ → 100% warm
30◦ < T < 40◦ → 50% warm
40◦ < T → 0% warm
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Fuzzy sets (main aim)
Mathematical treatment of uncertainty, ambiguity, subjectivity,...
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Fuzzy sets (main aim)
Mathematical treatment of uncertainty
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Fuzzy sets (main aim)
Mathematical treatment of uncertainty
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A real example in sociology
Limits of a metropolitan area
Joaquin Susino (University of Granada)
Eva Barrena (University of Sevilla)
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A real example in sociology
Limits of a metropolitan area
Joaquin Susino (University of Granada)
Eva Barrena (University of Sevilla)
Sevilla (membership 0 � 0.37 � 0.5 � 0.75 � 1)
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Applications in:
Fuzzy Logic
Patter Recognition
Knowledge Discovery
Data Mining
Database Management Systems
Automata Theory
Arti�cial Intelligence
...
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Origins: Zadeh and Rosenfeld
De�nition (Zadeh, 1965)
A fuzzy (sub)set of a set X is a map µ : X → [0, 1].
Set operations are made via pointwise lattice structure:I Intersection. (µ ∩ ν)(x) = µ(x) ∧ ν(x)I Union. (µ ∪ ν)(x) = µ(x) ∨ ν(x)I Complementary. µc(x) = 1− µ(x)I Subsets. µ ⊂ ν if and only if µ(x) ≤ ν(x)
Level cuts: µα = {x ∈ X | µ(x) ≥ α} for all α ∈ [0, 1].
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Origins: Zadeh and Rosenfeld
De�nition (Zadeh, 1965)
A fuzzy (sub)set of a set X is a map µ : X → [0, 1].
Set operations are made via pointwise lattice structure:I Intersection. (µ ∩ ν)(x) = µ(x) ∧ ν(x)I Union. (µ ∪ ν)(x) = µ(x) ∨ ν(x)I Complementary. µc(x) = 1− µ(x)I Subsets. µ ⊂ ν if and only if µ(x) ≤ ν(x)
Level cuts: µα = {x ∈ X | µ(x) ≥ α} for all α ∈ [0, 1].
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Origins: Zadeh and Rosenfeld
De�nition (Zadeh, 1965)
A fuzzy (sub)set of a set X is a map µ : X → [0, 1].
Set operations are made via pointwise lattice structure:I Intersection. (µ ∩ ν)(x) = µ(x) ∧ ν(x)I Union. (µ ∪ ν)(x) = µ(x) ∨ ν(x)I Complementary. µc(x) = 1− µ(x)I Subsets. µ ⊂ ν if and only if µ(x) ≤ ν(x)
Level cuts: µα = {x ∈ X | µ(x) ≥ α} for all α ∈ [0, 1].
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Origins: Zadeh and Rosenfeld
De�nition (Rosenfeld, 1971)
A fuzzy subgroup of a group G is a fuzzy subset H : G→ [0, 1] such that
H(xy) ≥ H(x) ∧H(y) and
H(x−1) ≥ H(x),
or equivalently
H(xy−1) ≥ H(x) ∧H(y).
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Fuzzy ideals
De�nition (Liu, 1982)
A fuzzy left (right resp. two-sided) ideal of a ring R is a fuzzy setI : R→ [0, 1] such that
I(x− y) = I(x) ∧ I(y),I(xy) ≥ I(y) (I(xy) ≥ I(x) resp. I(xy) ≥ I(x) ∨ I(y)).
Known facts:
I(1) ≤ I(x) ≤ I(0) for all x ∈ R.A fuzzy set I : R→ [0, 1] is a left (right resp. two-sided) ideal if and onlyif Iα is a left (right resp. two-sided) crisp ideal of R for allI(1) < α ≤ I(0).
Products:
Product of fuzzy subsets: (A ◦B)(x) =∨x=x1x2
(A(x1) ∧B(x2)).
Swamy and Swamy: IJ(x) =∨x=
∑i aibi
∧i(I(ai) ∧ J(bi)).
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Fuzzy ideals
De�nition (Liu, 1982)
A fuzzy left (right resp. two-sided) ideal of a ring R is a fuzzy setI : R→ [0, 1] such that
I(x− y) = I(x) ∧ I(y),I(xy) ≥ I(y) (I(xy) ≥ I(x) resp. I(xy) ≥ I(x) ∨ I(y)).
Known facts:
I(1) ≤ I(x) ≤ I(0) for all x ∈ R.A fuzzy set I : R→ [0, 1] is a left (right resp. two-sided) ideal if and onlyif Iα is a left (right resp. two-sided) crisp ideal of R for allI(1) < α ≤ I(0).
Products:
Product of fuzzy subsets: (A ◦B)(x) =∨x=x1x2
(A(x1) ∧B(x2)).
Swamy and Swamy: IJ(x) =∨x=
∑i aibi
∧i(I(ai) ∧ J(bi)).
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Index
1 Introduction and History
2 A survey on fuzzy prime ideals
3 Fuzzy primeness over noncommutative rings
4 Fuzzy semiprimeness and fuzzy prime radical
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Fuzzy primes revisionD0: xtys ≤ P ⇒ xt ≤ P or ys ≤ P
D0': 〈xt〉〈ys〉 ≤ P ⇒ xt ≤ P or ys ≤ P
D1: I ◦ J ≤ P ⇒ I ≤ P or J ≤ P one�sided versions of D1
D4: P (xy) = P (x) or P (xy) = P (y)
D2: Pα is prime for all P (1) < α ≤ P (0)
D3: P (xy) = P (0)⇒ P (x) = P (0) or P (y) = P (0)
Zahedi
Malik and Mordeson,Swamy and Swamy �4
Kumbhojkar and BapatCommut.
Zahedi
Kumbhojkar and Bapat �5
Commut.
Commut.
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Fuzzy primes revisionD0: xtys ≤ P ⇒ xt ≤ P or ys ≤ P
D0': 〈xt〉〈ys〉 ≤ P ⇒ xt ≤ P or ys ≤ P
D1: I ◦ J ≤ P ⇒ I ≤ P or J ≤ P one�sided versions of D1
D4: P (xy) = P (x) or P (xy) = P (y)
D2: Pα is prime for all P (1) < α ≤ P (0)
D3: P (xy) = P (0)⇒ P (x) = P (0) or P (y) = P (0)
Zahedi
Malik and Mordeson,Swamy and Swamy �4
Kumbhojkar and BapatCommut.
Zahedi
Kumbhojkar and Bapat �5
Commut.
Commut.
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Fuzzy primes revisionD0: xtys ≤ P ⇒ xt ≤ P or ys ≤ P
D0': 〈xt〉〈ys〉 ≤ P ⇒ xt ≤ P or ys ≤ P
D1: I ◦ J ≤ P ⇒ I ≤ P or J ≤ P one�sided versions of D1
D4: P (xy) = P (x) or P (xy) = P (y)
D2: Pα is prime for all P (1) < α ≤ P (0)
D3: P (xy) = P (0)⇒ P (x) = P (0) or P (y) = P (0)
Zahedi
Malik and Mordeson,Swamy and Swamy �4
Kumbhojkar and BapatCommut.
Zahedi
Kumbhojkar and Bapat �5
Commut.
Commut.
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Fuzzy primes revisionD0: xtys ≤ P ⇒ xt ≤ P or ys ≤ P
D0': 〈xt〉〈ys〉 ≤ P ⇒ xt ≤ P or ys ≤ P
D1: I ◦ J ≤ P ⇒ I ≤ P or J ≤ P one�sided versions of D1
D4: P (xy) = P (x) or P (xy) = P (y)
D2: Pα is prime for all P (1) < α ≤ P (0)
D3: P (xy) = P (0)⇒ P (x) = P (0) or P (y) = P (0)
Zahedi
Malik and Mordeson,Swamy and Swamy �4
Kumbhojkar and BapatCommut.
Zahedi
Kumbhojkar and Bapat �5
Commut.
Commut.
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Fuzzy primes revisionD0: xtys ≤ P ⇒ xt ≤ P or ys ≤ P
D0': 〈xt〉〈ys〉 ≤ P ⇒ xt ≤ P or ys ≤ P
D1: I ◦ J ≤ P ⇒ I ≤ P or J ≤ P one�sided versions of D1
D4: P (xy) = P (x) or P (xy) = P (y)
D2: Pα is prime for all P (1) < α ≤ P (0)
D3: P (xy) = P (0)⇒ P (x) = P (0) or P (y) = P (0)
Zahedi
Malik and Mordeson,Swamy and Swamy �4
Kumbhojkar and BapatCommut.
Zahedi
Kumbhojkar and Bapat �5
Commut.
Commut.
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Fuzzy primes revisionD0: xtys ≤ P ⇒ xt ≤ P or ys ≤ P
D0': 〈xt〉〈ys〉 ≤ P ⇒ xt ≤ P or ys ≤ P
D1: I ◦ J ≤ P ⇒ I ≤ P or J ≤ P one�sided versions of D1
D4: P (xy) = P (x) or P (xy) = P (y)
D2: Pα is prime for all P (1) < α ≤ P (0)
D3: P (xy) = P (0)⇒ P (x) = P (0) or P (y) = P (0)
Zahedi
Malik and Mordeson,Swamy and Swamy �4
Kumbhojkar and BapatCommut.
Zahedi
Kumbhojkar and Bapat �5
Commut.
Commut.
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Fuzzy primes revisionD0: xtys ≤ P ⇒ xt ≤ P or ys ≤ P
D0': 〈xt〉〈ys〉 ≤ P ⇒ xt ≤ P or ys ≤ P
D1: I ◦ J ≤ P ⇒ I ≤ P or J ≤ P one�sided versions of D1
D4: P (xy) = P (x) or P (xy) = P (y)
D2: Pα is prime for all P (1) < α ≤ P (0)
D3: P (xy) = P (0)⇒ P (x) = P (0) or P (y) = P (0)
Zahedi
Malik and Mordeson,Swamy and Swamy �4
Kumbhojkar and BapatCommut.
Zahedi
Kumbhojkar and Bapat �5
Commut.
Commut.
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Some Lemmas
Lemma (Malik and Mordeson, Swamy and Swamy)
A fuzzy ideal P : R→ [0, 1] is D1-prime if and only if P has the form
P (x) =
{1 if x ∈ Q,t otherwise,
where Q is a crisp prime ideal of R and 0 ≤ t < 1.
Lemma (Kumbhojkar and Bapat)
A fuzzy ideal P is D4-prime if and only if each level cut Pα is completely
prime for all P (0) ≥ α > P (1).
�25
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Some examples
Example
Let R be the ring of 2× 2 matrices over the real numbers. Consider the fuzzyideal
P (x) =
{1 if x is the zero matrix,0 otherwise.
The zero ideal is prime, therefore P is D2-prime and D1-prime. Nevertheless
P (( 0 10 0 ) (
0 10 0 )) = P (( 0 0
0 0 )) = 1 whilst P (( 0 10 0 )) = 0,
so P is not D4-prime.
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More examples
Example
R =M2(R). Let P be the fuzzy ideal given by
P (x) =
{1 if x is the zero matrix,0 otherwise
and x1 be the singleton of the element x = ( 0 10 0 ) . On the one hand,
x1x1(z) =∨
z=∑
i zi1z
i2
∧i
(x1(zi1) ∧ x1(zi2)) 6= 0⇒ zi1 = x, zi2 = x⇒ z = ( 0 0
0 0 )
and x1x1(( 0 00 0 )) = 1. Then x1x1 = P and 〈x1x1〉 = P . On the other hand,
〈x1〉 is a non-zero fuzzy ideal with 〈x1〉(x) = 1. Since R is a simple ring,〈x1〉 = R and, consequently, 〈x1〉 ◦ 〈x1〉 = R * P .R is a prime ring hence, by Lemma 4, P is D1-prime. Since x1 � P andx1x1 = P , it follows that P is not D0-prime. This contradicts some results of(Kumbhojkar and Bapat, 1993) and (Malik and Mordeson, 1992) if we omitthe commutativity condition on the ring.
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Index
1 Introduction and History
2 A survey on fuzzy prime ideals
3 Fuzzy primeness over noncommutative rings
4 Fuzzy semiprimeness and fuzzy prime radical
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Our proposal of fuzzy prime ideals
De�nition
Let R be an arbitrary ring with unity. A non-constant fuzzy idealP : R→ [0, 1] is said to be prime if, for any x, y ∈ R,
∧P (xRy) = P (x)∨P (y).
Theorem
Let R be an arbitrary ring with unity and P : R→ [0, 1] be a non-constant
fuzzy ideal of R. The following conditions are equivalent:
a) P is prime.
b) Pα is prime for all P (0) ≥ α > P (1).
Moreover, if R is commutative, any of these statements is equivalent to Pbeing D4-prime.
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Equivalence of fuzzy sets (Kumbhojkar and Bapat)
Two fuzzy sets I and J are equivalent if, for any x, y ∈ R,I(x) > I(y) ⇐⇒ J(x) > J(y).
The fuzzy ideals equivalent to the zero ideal are of the form Os<t(0) = tand Os<t(x) = s for x 6= 0, where 0 ≤ s < t ≤ 1.
Corollary
Let R be an arbitrary ring with unity. The following statements are equivalent:
a) R is prime.
b) There exist 0 ≤ s < t ≤ 1 such that Os<t is a prime fuzzy ideal.
c) For all 0 ≤ s < t ≤ 1, Os<t is a prime fuzzy ideal.
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Some di�erences with crisp theory
LemmaAny prime fuzzy ideal contains properly another prime fuzzy ideal.
De�nitionA prime fuzzy ideal P is said to be minimal if it is equivalent to thecharacteristic map of a minimal prime ideal.
Proposition
Any prime fuzzy ideal contains a minimal prime fuzzy ideal.
Corollary
Let R be a noetherian ring. The number of equivalence classes of minimal
prime fuzzy ideals is �nite.
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Index
1 Introduction and History
2 A survey on fuzzy prime ideals
3 Fuzzy primeness over noncommutative rings
4 Fuzzy semiprimeness and fuzzy prime radical
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Fuzzy semiprimes revision
D0': 〈xt〉2 ≤ P ⇒ xt ≤ P
D1: I2 ≤ P ⇒ I ≤ P one�sided versions of D1
D4: P (x2) = P (x)
D2: Pα is semiprime for all P (1) < α ≤ P (0)
Commut.
Commut.
Commut.
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Our proposal on fuzzy semiprimes
De�nitionLet R be an arbitrary ring with unity. A non-constant fuzzy idealP : R→ [0, 1] is said to be semiprime if
∧P (xRx) = P (x) for all x ∈ R.
Proposition
Let R be an arbitrary ring with unity and P : R→ [0, 1] be a non-constant
fuzzy ideal of R. The following conditions are equivalent:
a) P is semiprime.
b) Pα is semiprime for all P (0) ≥ α > P (1).
Moreover, if R is commutative, any of these statements is equivalent to Pbeing D4-semiprime.
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Expected connection of primes and semiprimes
Theorem
A fuzzy ideal is semiprime if and only if it is the intersection of prime fuzzy
ideals.
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Sketch of the proof, only if
∧(⋂j∈J
Pj)(xRx) =∧r∈R
(⋂j∈J
Pj)(xrx)
=∧r∈R
∧j∈J
Pj(xrx)
=∧j∈J
∧r∈R
Pj(xrx)
=∧j∈J
∧Pj(xRx)
†=∧j∈J
Pj(x)
= (⋂j∈J
Pj)(x),
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Sketch of the proof, ifLet P be a semiprime fuzzy ideal.
C = {prime fuzzy ideals Q such that P ≤ Q}.
By Zorn's Lemma ∃M maximal (hence prime) ideal of R such thatM ⊇ P ∗ = {x ∈ R such that P (x) > P (1)} The fuzzy ideal
H(x) =
{P (0) if x ∈M ,P (1) otherwise.
satis�es H ∈ C.
Assume ∃x ∈ R such that P (x) < (⋂Q)(x) =
∧Q(x). If P (x) = P (0), then
H(x) = P (0) = P (x) < (⋂Q)(x) !!!
Else let t ∈ (0, 1) such that P (x) < t <∧Q(x) and t < P (0). Now, Pt is
semiprime and x /∈ Pt. As a consequence of Zorn's Lemma there exists a primeideal M with Pt ⊆M and x /∈M . We de�ne
I(z) =
{P (0) if z ∈M ,t otherwise.
We can check than I ∈ C and I(x) = t < (⋂Q)(x) !!!.
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Sketch of the proof, ifLet P be a semiprime fuzzy ideal.
C = {prime fuzzy ideals Q such that P ≤ Q}.
By Zorn's Lemma ∃M maximal (hence prime) ideal of R such thatM ⊇ P ∗ = {x ∈ R such that P (x) > P (1)} The fuzzy ideal
H(x) =
{P (0) if x ∈M ,P (1) otherwise.
satis�es H ∈ C.Assume ∃x ∈ R such that P (x) < (
⋂Q)(x) =
∧Q(x). If P (x) = P (0), then
H(x) = P (0) = P (x) < (⋂Q)(x) !!!
Else let t ∈ (0, 1) such that P (x) < t <∧Q(x) and t < P (0). Now, Pt is
semiprime and x /∈ Pt. As a consequence of Zorn's Lemma there exists a primeideal M with Pt ⊆M and x /∈M . We de�ne
I(z) =
{P (0) if z ∈M ,t otherwise.
We can check than I ∈ C and I(x) = t < (⋂Q)(x) !!!.
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Sketch of the proof, ifLet P be a semiprime fuzzy ideal.
C = {prime fuzzy ideals Q such that P ≤ Q}.
By Zorn's Lemma ∃M maximal (hence prime) ideal of R such thatM ⊇ P ∗ = {x ∈ R such that P (x) > P (1)} The fuzzy ideal
H(x) =
{P (0) if x ∈M ,P (1) otherwise.
satis�es H ∈ C.Assume ∃x ∈ R such that P (x) < (
⋂Q)(x) =
∧Q(x). If P (x) = P (0), then
H(x) = P (0) = P (x) < (⋂Q)(x) !!!
Else let t ∈ (0, 1) such that P (x) < t <∧Q(x) and t < P (0). Now, Pt is
semiprime and x /∈ Pt. As a consequence of Zorn's Lemma there exists a primeideal M with Pt ⊆M and x /∈M . We de�ne
I(z) =
{P (0) if z ∈M ,t otherwise.
We can check than I ∈ C and I(x) = t < (⋂Q)(x) !!!.
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Going to the Fuzzy Radical
Corollary
Let R be a ring with unity and I be a non-constant fuzzy ideal over R. The
following fuzzy ideals coincide:
i) The intersection F1 of all semiprime fuzzy ideals containing I.
ii) The intersection F2 of all prime fuzzy ideals containing I.
iii) The fuzzy ideal F3 given by F3(x) =∨{t ∈ [0, 1] such that x ∈ Rad(It)}.
This fuzzy ideal is called the fuzzy prime radical and denoted by FRad(I).
Corollary
P is a semiprime fuzzy ideal if and only if FRad(P ) = P .
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Going to the Fuzzy Radical
Corollary
Let R be a ring with unity and I be a non-constant fuzzy ideal over R. The
following fuzzy ideals coincide:
i) The intersection F1 of all semiprime fuzzy ideals containing I.
ii) The intersection F2 of all prime fuzzy ideals containing I.
iii) The fuzzy ideal F3 given by F3(x) =∨{t ∈ [0, 1] such that x ∈ Rad(It)}.
This fuzzy ideal is called the fuzzy prime radical and denoted by FRad(I).
Corollary
P is a semiprime fuzzy ideal if and only if FRad(P ) = P .
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Properties of the Fuzzy Radical
Proposition
Let I and J be non-constant fuzzy ideals over R. The following statements
hold:
i) FRad(FRad(I)) = FRad(I).
ii) Rad(R/FRad(I)) = 0.
iii) If I ≤ J then FRad(I) ≤ FRad(J).
iv) FRad(I ∩ J) = FRad(I) ∩ FRad(J).
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Thanks for your attention!
Ups, just two more slides!
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Thanks for your attention!
Ups, just two more slides!
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