zDECSAI, Universidad de Granadagnavarro/files/fuzzy_slides.pdf · 2015-09-09 · Prime fuzzy ideals over noncommutative rings F. J. Lobillo y, O. Cortadellas and G. Naarrov z yDepartamento

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

Gabriel Navarro (UGR) Prime fuzzy ideals XIJTA 2012, May 2012 1 / 34

Work published in:Fuzzy Sets and Systems, 199 (2012) 108�120


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


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!





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!





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






Mathematical treatment of uncertainty



Mathematical treatment of uncertainty


A real example in sociology

Limits of a metropolitan area

Joaquin Susino (University of Granada)

Eva Barrena (University of Sevilla)


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)


Applications in:

Fuzzy Logic

Patter Recognition

Knowledge Discovery

Data Mining

Database Management Systems

Automata Theory

Arti�cial Intelligence

...


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].















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).


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)).


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)).


Index






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.



D0': 〈xt〉〈ys〉 ≤ P ⇒ xt ≤ P or ys ≤ P


D4: P (xy) = P (x) or P (xy) = P (y)


D3: P (xy) = P (0)⇒ P (x) = P (0) or P (y) = P (0)

Zahedi



Zahedi


Commut.

Commut.



D0': 〈xt〉〈ys〉 ≤ P ⇒ xt ≤ P or ys ≤ P


D4: P (xy) = P (x) or P (xy) = P (y)


D3: P (xy) = P (0)⇒ P (x) = P (0) or P (y) = P (0)

Zahedi



Zahedi


Commut.

Commut.



D0': 〈xt〉〈ys〉 ≤ P ⇒ xt ≤ P or ys ≤ P


D4: P (xy) = P (x) or P (xy) = P (y)


D3: P (xy) = P (0)⇒ P (x) = P (0) or P (y) = P (0)

Zahedi



Zahedi


Commut.

Commut.



D0': 〈xt〉〈ys〉 ≤ P ⇒ xt ≤ P or ys ≤ P


D4: P (xy) = P (x) or P (xy) = P (y)


D3: P (xy) = P (0)⇒ P (x) = P (0) or P (y) = P (0)

Zahedi



Zahedi


Commut.

Commut.



D0': 〈xt〉〈ys〉 ≤ P ⇒ xt ≤ P or ys ≤ P


D4: P (xy) = P (x) or P (xy) = P (y)


D3: P (xy) = P (0)⇒ P (x) = P (0) or P (y) = P (0)

Zahedi



Zahedi


Commut.

Commut.



D0': 〈xt〉〈ys〉 ≤ P ⇒ xt ≤ P or ys ≤ P


D4: P (xy) = P (x) or P (xy) = P (y)


D3: P (xy) = P (0)⇒ P (x) = P (0) or P (y) = P (0)

Zahedi



Zahedi


Commut.

Commut.


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


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.


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.


Index






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.


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.


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.


Index






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.


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.


Expected connection of primes and semiprimes

Theorem

A fuzzy ideal is semiprime if and only if it is the intersection of prime fuzzy

ideals.


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),


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) !!!.





H(x) =


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) !!!



I(z) =







H(x) =


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) !!!



I(z) =




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 .


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 .


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).


Thanks for your attention!

Ups, just two more slides!


Thanks for your attention!

Ups, just two more slides!




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zDECSAI, Universidad de Granadagnavarro/files/fuzzy_slides.pdf · 2015-09-09 · Prime fuzzy ideals over noncommutative rings F. J. Lobillo y, O. Cortadellas and G. Naarrov z yDepartamento