Let R be a commutative ring with unity. In this paper we introduce and study Fuzzy Quasi regular ring as generalizations of (ordinary) Quasi regular ring. We give some basic properties about these concepts.

1. Integrated Intelligent Research (IIR) International Journal of Data Mining Techniques and Applications
Volume: 06 Issue: 01 June 2017, Page No.36-38
ISSN: 2278-2419
36
Fuzzy Quasi Regular Ring
Shrooq Bahjat Smeein
Lecturer - Department of IT (Math Section), Shinas college of Technology. Oman
Email: Shrooq.Smeein@shct.edu.om
Abstract - Let R be a commutative ring with unity. In this
paper we introduce and study Fuzzy Quasi regular ring as
generalizations of (ordinary) Quasi regular ring. We give some
basic properties about these concepts.
Keywords: Fuzzy set ,Fuzzy ring , Quasi regular ring ,Fuzzy
maximal ideal
I. INTRODUCTION
The notation of Fuzzy subset was introduced by L.A.Zadeh [1].
His seminar paper in 1965 has opened up new insights and
applications in a wide range of scientific fields. Liu [2] has
studies Fuzzy ideals and Fuzzy ring. Regular rings were
presented by Skornyakov [3] as a generalization this to concept
(Nicholson) gave the concept of Quasi regular ring [4]. We
present Fuzzy Quasi regular ring by extending the notation of
(ordinary) Quasi regular ring. We give some basic results
about Fuzzy Quasi regular ring. Also we study the
holomorphic image of Fuzzy Quasi regular ring.
II. FUZZY QUASI REGULAR RING
1.1 DEFINITION [1]
Let S be a non-empty set and I be the closed interval
[0,1] of the real line (real numbers). A Fuzzy set A in
S (a Fuzzy subset of S) is a function from S into I.
1.2 Definition [2]
Let Xt: S  [0, 1] be a Fuzzy set in S, where xS
t[0,1] defined by: Xt(y) =t if x=y, and Xt(y) = 0
if xy yS. Xt is called a Fuzzy singleton or
Fuzzy point on S.
1.3 Remark [1]
The following properties of level subsets hold for each t(0,1]
1. (AB)t=At  Bt
2. A=B if and only if At=Bt, for all t(0,1].
1.4 Definition [5]
Let (R,+,) be a ring and let X be a Fuzzy set in ٌ R. Then X is
called a Fuzzy ring in ring (R,+,) if and only if, for each x, y 
R
1. X(x+y)  min{X(x), X(y)}
2. X(x) = X(– x)
3. X(xy)  min{X(x), X(y)}.
1.5 Definition [6]
A Fuzzy subset X of a ring R is called a Fuzzy ideal of R, if for
each x, y  R
1. X(x–y)  min{X(x), X(y)}
2. X(xy)  max{X(x), X(y)}.
1.6 Definition [7]
A ring R is said to be Quasi regular if for R
a there is
Ra
e
e 

2
such that )
(
)
1
( R
J
e
a 
 .
(For all R
a , there is R
x such that xa
xa 
2
)
( and
)
(R
J
axa
a 
 ).
We fuzzified this concept as follows.
1.7 Definition
Let X be Fuzzy ring, X is called Fuzzy Quasi regular
If and only if ,
, X
e
X
a k
t 


 k
k e
e 
2
Such that )
(
)
1
( )
0
( X
J
F
e
a k
x
t 

 .
( X
x
X
a k
t 


 , such that
2
)
( k
t
k
t x
a
x
a  ,
)
(
2
X
J
F
x
a
a k
t
t 

 )
1.8 Proposition
Let X be a Fuzzy ring )
(
, X
J
F
a
X
a t
t 

 then
k
t r
a

1 invertible singleton in X.
Proof: Let 


 t
t a
X
J
F
a )
( any maximal Fuzzy
ideal of X
ideal of X
Suppose k
t
x r
a

)
0
(
1 is not invertible singleton, then there
is a Maximal Fuzzy ideal A of X Such that 

 k
t
x r
a
)
0
(
1
A
 .
Hence k
t
x r
a

)
0
(
1 A
 ,but A
at  implies that A
r
a k
t  ,
hence A
x 
)
0
(
1 .
This implies X
A  because for any X
bs  , s
x
s b
b )
0
(
1

Which implies A
bs  since A
x 
)
0
(
1 .
Thus A
X  which is contradiction.
1.9 Proposition
Let X be a Fuzzy ring then )
(
))
(
( t
t X
J
X
J
F 
 ,
]
1
,
0
[

t .
Proof: Let t
X
J
F
a ))
(
( 
 , Hence )
(X
J
F
at 

To prove )
( t
X
J
a
We must prove ,
t
X
r
 ar

1 invertible in t
X
Since X
r
X
a t
t
t 



2. Integrated Intelligent Research (IIR) International Journal of Data Mining Techniques and Applications
Volume: 06 Issue: 01 June 2017, Page No.36-38
ISSN: 2278-2419
37
Since X
r
X
J
F
a t
t 

 ),
( , we have t
t
x a
r

)
0
(
1
invertible (singleton) in X
Implies that t
ra)
1
(  invertible singleton in X
Then X
bl 
 such that )
0
(
1
)
1
( x
l
t b
ra 

Implies that )
0
(
1
]
)
1
[( x
b
ra 
 
)
0
(
x

  and 1
)
1
( 
 b
ra
)
1
( ra

 inv.elem.in t
X
)
( t
X
J
a

The following result explains the relationship between
Fuzzy Quasi regular ring and its level.
1.10 Corollary
Let X be a Fuzzy Quasi regular ring, then t
X is Quasi-regular
ring, )
1
,
0
[

t .
Proof: Let t
X
a , then X
at  .But X is
Fuzzy Quasi regular ring
So X
rk 
 such that )
(
2
X
J
F
r
a
a k
t
t 


Then )
(
)
( 2
X
J
F
r
a
a 

  , }
,
min{ k
t


)
(
))
(
(
2

 X
J
X
J
F
r
a
a 


 ,
Implies that )
(
2

X
J
r
a
a 

Since }
,
min{ k
t

 , so t

 or k


If t

 , then )
(
2
t
X
J
r
a
a 
 and t
X
is quasi regular ring
If k

 , then k

 and so 
X
X
X k
t 

)
(
2
t
X
J
r
a
a 

 and so t
X is Quasi regular ring
If k

 ,then k< t so 
X
X
X k
t 

{ maximal ideal in Xt } { maximal ideal in 
X }
Thus )
(
2
t
X
J
r
a
a 
 and so Xt is Quasi regular ring
1.11 Proposition
X is Fuzzy regular ring if and only if X is Fuzzy Quasi regular
ring and )
0
(
)
( x
O
X
J
F 

Proof: Let ,
X
at  since X is Fuzzy regular
X
rk 

 Suchthat
}
,
min{
),
(
)
0
(
2
k
t
X
J
F
O
O
r
a
a x
k
t
t 




 

Then X is Fuzzy Quasi regular ring
To prove )
0
(
)
( x
O
X
J
F 

Let )
(X
J
F
at 
 , since X is Fuzzy regular
X
rk 

 Such that )
0
(
2
x
k
t
t O
r
a
a 

)
0
(
)
0
( )
1
( x
k
t
x
t O
r
a
a 


By k
t
x
t r
a
X
J
F
a 


 )
0
(
1
)
(
( invertible)
X
bl 
 such that )
0
(
)
0
( 1
)
1
( x
l
k
t
x b
r
a 

Then )
0
(
)
0
( )
1
( x
l
k
t
x O
b
r
a 

)
0
(
)
0
(
)
0
(
1 x
t
t
x
x
t O
O
a
O
a 



Then )
0
(
)
( x
O
X
J
F 

1.12 Proposition
In a Fuzzy Quasi regular ring every Fuzzy non unit is a Fuzzy
zero divisor
Proof: Let X is a Fuzzy Quasi regular ring then t
X is Quasi
regular ring
Let X
at  not unit, then t
X
a (a is non unit)
)
(
2
t
X
J
r
a
a 


r
a
a 2
1 

 Invertible element
1.13 Theorem
Let X be a Fuzzy Quasi regular ring, then for every
Fuzzy ideal of X
)
(
)
( K
I
x
J
F
K
I
K
I 


 


Proof: Since K
I
K
I
X
J
F 

 
 )
(
)
(
and K
I
K
I 
 
K
I
K
I
X
J
F
K
I 


 


 )
(
)
(
Let K
I
at 
 ,since X is Fuzzy Quasi regular ring
X
xk 

 , )
(
2
X
J
F
x
a
a k
t
t 


 ,
2
)
( k
t
k
t x
a
x
a  K
I
x
a
a k
t
t 


 2
)
(
)
(
2
K
I
X
J
F
x
a
a k
t
t 





Let l
k
t
t b
x
a
a 
 2
, )
(
)
( K
I
x
J
F
bl 



l
k
t
t b
x
a
a 

 2
, it is clear K
I
x
a k
t 

2
)
(
)
(
)
(
)
(
K
I
X
J
F
K
I
K
I
K
I
X
J
F
K
I
at















1.14 Corollary
Let X be a Fuzzy Quasi regular ring, then for every fuzzy ideal
of X
I
x
J
F
I
I 
)
(
2



Proof: If K
I  ,then K
X
J
F
K
K 
)
(
2



)
(
)
(
)
( 2
K
I
X
J
F
K
I
K
I 


 

 ,
K
I
K
I 
 
2
)
(

3. Integrated Intelligent Research (IIR) International Journal of Data Mining Techniques and Applications
Volume: 06 Issue: 01 June 2017, Page No.36-38
ISSN: 2278-2419
38
)
(
)
(
)
(
)
(
)
(
)
(
K
I
X
J
F
K
I
K
I
K
I
K
I
X
J
F
K
I
K
I
X
J
F
K
I
K
I























Then I
x
J
F
I
I 
)
(
2



1.15 Corollary
Let X be a Fuzzy Quasi regular ring then for every
X
b
a h
t 
,










 h
t
t
l b
a
X
J
F
ab
a
b 

 )
(
)
( 
1.16 Proposition
If A is Fuzzy ideal such that X
a
A t 

 A
at 
 ,
then A is maximal Fuzzy ideal
Proof: Suppose B
 Fuzzy ideal such that X
B
A 

Then B
at 
 and X
B
a
A
A
A
a t
t 






X
B
X
B
X 



 , then A is Fuzzy maximal
1.17 Lemma
Let Y
X
f 
: , f is homomorphism, then
)
(
))
(
( Y
J
F
X
J
F
f 


Proof: )
(
))
(
( i
i
L
f
X
J
F
f



 
( l
L is Fuzzy maximal ideal in X )
( i
i
L
f


  )
But )
( i
L
f is Fuzzy maximal ideal in Y )
(Y
J
F 
 
1.18 Proposition
Let Y
X
f 
: epimorphism. If X is Quasi regular Fuzzy
ring then Y is Quasi regular Fuzzy ring
Proof: Let Y
yt  , since f is onto R
x


such that ,
)
( y
x
f  hence )
( t
t x
f
y 
But )
(
)
(
2
X
J
F
r
x
x
f k
t
t 

 for some X
rk 
 )
(
))
(
(
)
(
2
Y
J
F
X
J
F
f
r
x
x
f k
t
t 




)
(
))
(
)(
(
)
( 2
Y
J
F
x
f
r
f
x
f t
k
t 



)
(
2
/
Y
J
F
y
r
y t
t
t 



Then Y is Quasi regular Fuzzy ring
Recommendations
The extension to this can be given to find the Inverse Image for
𝑓: 𝑋 → 𝑌 Quasi regular Fuzzy ring, Other researchers must
conduct similar studies related to quasi regular Fuzzy ring and
to further validate the claims of this study
References
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8, 338-353.
[2] Liu, W.J., (1982), Fuzzy Invariant Subgroups and Fuzzy
Ideals, Fuzzy Sets and Systems, 8, 133-139.
[3] L.A. Skornyakov (2001), "Regular ring (in the sense of
von Neumann)", in Hazewinkel, E cyclopedia of
Mathematics, Springer, ISBN 978-1-55608-010-4
[4] Nicholson, w. Keith. (1973): "Rings whose Elements
are Quasi-Regular or regular." Equations Mathematica
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[5] Nanda, S., (1989), Fuzzy Modules Over Fuzzy Rings,
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[7] Nuhad S.Al Mathafar; (1993) Z-regular Modules, M.Sc.
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