International Journal of Engineering Research and Development

1. International Journal of Engineering Research and Development
e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com
Volume 10, Issue 7 (July 2014), PP.61-66
Interval Valued Intuitionistic Fuzzy Subrings of A Ring
1M.G.Somasundara Moorthy & 2K.Arjunan
1. Department of Mathematics, Mookambigai College of Engineering, Keeranur, Tamilnadu, India.
2. Department of Mathematics, H.H.The Rajahs College, Pudukkottai – 622001, Tamilnadu, India
Abstract:- In this paper, we study some of the properties of interval valued intuitionistic fuzzy subring of a ring
and prove some results on these.
2000 AMS SUBJECT CLASSIFICATION: 03F55, 08A72, 20N25.
Keywords:- Interval valued fuzzy subset, interval valued fuzzy subring, interval valued intuitionistic fuzzy
subset, interval valued intuitionistic fuzzy subring.
I. INTRODUCTION
Interval-valued fuzzy sets were introduced independently by Zadeh [13], Grattan-Guiness [5], Jahn [7],
in the seventies, in the same year. An interval valued fuzzy set (IVF) is defined by an interval-valued
membership function. Jun.Y.B and Kin.K.H[8] defined an interval valued fuzzy R-subgroups of nearrings.
Solairaju.A and Nagarajan.R[10] defined the charactarization of interval valued Anti fuzzy Left h-ideals over
Hemirings. M.G.Somasundara Moorthy and K. Arjunan[11] have defined an interval valued fuzzy subring of a
ring under homomorphism. We introduce the concept of interval valued intuitionistic fuzzy subring of a ring and
established some results.
1.PRELIMINARIES:
1.1 Definition: Let X be any nonempty set. A mapping [M] : X  D[0, 1] is called an interval valued fuzzy
subset (briefly, IVFS ) of X, where D[0,1] denotes the family of all closed subintervals of [0,1] and [M](x) =
[M(x), M+(x)], for all x in X, where M and M+ are fuzzy subsets of X such that M(x) ≤ M+(x), for all x in X.
Thus M(x) is an interval (a closed subset of [0,1] ) and not a number from the interval [0,1] as in the case of
fuzzy subset. Note that [0] = [0, 0] and [1] = [1, 1].
1.2 Definition: Let ( R, +, ∙ ) be a ring. An interval valued fuzzy subset [M] of R is said to be an interval
valued fuzzy subring(IVFSR) of R if the following conditions are satisfied:
(i) [M](x+y)  rmin { [M](x), [M](y) },
(ii) [M](x)  [M](x),
(iii) [M](xy)  rmin { [M](x), [M](y) }, for all x and y in R.
1.3 Definition: An interval valued intuitionistic fuzzy subset (IVIFS) [A] in X is defined as an object of the
form [A] = { < x, [A](x), [A](x) > / x in X}, where [A] : X  D[0, 1] and [A] : X  D[0, 1]
define the degree of membership and the degree of non-membership of the element xX respectively and for
every xX satisfying [A]
61
+(x) + [A]
+(x)  1.
1.4 Definition: Let ( R, +, .) be a ring. An interval valued intuitionistic fuzzy subset [A] of R is said to be an
interval valued intuitionistic fuzzy subring of R if it satisfies the following axioms:
(i) [A](x y) ≥ rmin {A](x), A](y)} = [min {+(+([[[A]
(x), [A]
(y)}, min{[A]
x), [A]
y)}]
(ii) [A](xy) ≥ rmin {[A](x), [A](y)} = [ min {[A]
(x), [A]
(y)}, min{[A]
+(x), [A]
+(y)}]
(iii) [A](xy) ≤ rmax{[A](x), [A](y)} = [max {[A]
(x), [A]
(y)}, max{[A]
+(x), [A]
+(y)}]
(iv) [A](xy) ≤ rmax {[A](x), [A](y)} = [ max{[A]
(x), [A]
(y)}, max{[A]
+(x), [A]
+(y)}], for all x and y in R.
1.5 Definition: Let X and X be any two sets. Let f : X  X be any function and [A] be an interval valued
intuitionistic fuzzy subset in X, [V] be an interval valued intuitionistic fuzzy subset in f(X) = X, defined by
[V](y) = sup
r
x f y  
( ) 1
( ) 1 r
x f y  
[A](x) and [V](y) = inf
[A](x), for all x in X and y in X. [A] is called a preimage of [V]
under f and is denoted by f-1([V]).
1.6 Definition: Let [A] be an interval valued intuitionistic fuzzy subring of a ring R and a in R. Then the pseudo
interval valued intuitionistic fuzzy coset (aA)p is defined by ((a[A])p )(x) = p(a)[A](x) and ((a[A])p)(x) =
p(a)[A](x), for every x in R and for some p in P.
1.7 Definition: Let [A] and [B] be interval valued intuitionistic fuzzy subsets of sets G and H, respectively. The
product of [A] and [B], denoted by [A]×[B], is defined as [A]×[B] = { (x, y), [A]×[B](x,y), [A]×[B](x,y)  / for all

2. Interval Valued Intuitionistic Fuzzy Subrings of A Ring
x in G and y in H }, where [A]×[B](x, y) = rmin { [A](x), [B](y) } and [A]×[B](x, y) = rmax { [A](x),
[B](y) }.
1.8 Definition: Let [A] be an interval valued intuitionistic fuzzy subset in a set S. The strongest interval
valued intuitionistic fuzzy relation on S, that is an interval valued intuitionistic fuzzy relation on [A] is [V]
given by [V](x, y) = rmin { [A](x), [A](y) } and [V](x, y) = rmax{ [A](x), [A](y) }, for all x, y in S.
II. SOME PROPERTIES
2.1 Theorem: Intersection of any two interval valued intuitionistic fuzzy subrings of a ring R is an interval
valued intuitionistic fuzzy subring of R.
Proof: Let [A] and [B] be any two interval valued intuitionistic fuzzy subrings of a ring R and x, y in R. Let [A]
= {(x, [A](x), [A](x))/ xR} and [B] = { (x, [B](x), [B](x) ) / xR} and also let [C] = [A][B] = { (x, [C](x),
[C](x) ) / xR}, where [C](x) = rmin {[A](x), [B](x) } and [C](x) = rmax {[A](x), [B](x) }. Now, [C](xy) =
rmin {[A](xy), [B](xy) }≥ rmin{ rmin { [A](x), [A](y) }, rmin { [B](x), [B](y) }} = rmin {rmin{ [A](x),
[B](x) }, rmin { [A](y), [B](y) }} = rmin { [C](x), [C](y) }. Therefore, [C](xy) ≥ rmin { [C](x), [C](y) }, for
all x, y in R. And, [C](xy) = rmin { [A](xy), [B](xy) } ≥ rmin { rmin{ [A](x), [A](y) }, rmin { [B](x), [B](y)
}} = rmin{ rmin {[A](x), [B](x) }, rmin { [A](y), [B](y) }} = rmin{ [C](x), [C](y) }. Therefore, [C](xy) ≥
rmin{ [C](x), [C](y) }, for all x, y in R. Now, [C](xy) = rmax { [A](xy), [B](xy)} ≤ rmax{rmax {[A](x),
[A](y) }, rmax { [B](x), [B](y) }} = rmax{ rmax{[A](x), [B](x)}, rmax { [A](y), [B](y)}} = rmax { [C](x),
[C](y) }. Therefore, [C](xy) ≤ rmax { [C](x), [C](y) }, for all x, y in R. And, [C](xy) = rmax { [A](xy),
[B](xy) } ≤ rmax { rmax {[A](x), [A](y) }, rmax { [B](x), [B](y) }}= rmax{ rmax { [A](x), [B](x) }, rmax
{[A](y), [B](y) }} = rmax{ [C](x), [C](y)}. Therefore, [C](xy) ≤ rmax{ [C](x), [C](y) }, for all x, y in R.
Therefore [C] is an interval valued intuitionistic fuzzy subring of R. Hence the intersection of any two interval
valued intuitionistic fuzzy subrings of a ring R is an interval valued intuitionistic fuzzy subring of R.
2.2 Theorem: The intersection of a family of interval valued intuitionistic fuzzy subrings of a ring R is an
interval valued intuitionistic fuzzy subring of R.
Proof: Let { [V]i : iI} be a family of interval valued intuitionistic fuzzy subrings of a ring R and let [A] =
inf [V]i(xy) ≥ r
inf [V]i(y) } = rmin {[A](x), [A](y) }. Therefore, [A](xy) ≥ rmin{
sup [V]i(xy) ≤ r
sup [V]i(y) } = rmax{ [A](x), [A](y) }. Therefore, [A](x-y) ≤ rmax{ [A](x), [A](y) }, for all x, y
sup rmax{ [V]i(x), [V]i(y) } = rmax {r
62
i
i I
[V]
 
. Let x and y in R. Then, [A](xy) = r
iI
inf rmin {[V]i(x), [V]i(y) } = rmin {
iI
r
inf [V]i(x), r
iI
inf [V]i(y) }= rmin{[A](x), [A](y)}.
iI
Therefore, [A](xy) ≥ rmin { [A](x), [A](y) }, for all x, y in R. And, [A](xy) = r
inf [V]i(xy) ≥ r
iI
inf rmin{
iI
[V]i(x), [V]i(y) } = rmin{ r
inf [V]i(x), r
iI
iI
[A](x), [A](y) }, for all x, y in R. Now, [A](xy) = r
iI
sup rmax { [V]i(x), [V]i(y) } = rmax {
iI
r
sup [V]i(x), r
iI
iI
in R. And, [A](xy) = r
sup [V]i(xy) ≤ r
iI
iI
sup [V]i(x), r
iI
sup [V]i(y) } =
iI
rmax{ [A](x), [A](y) }. Therefore, [A](xy) ≤ rmax {[A](x), [A](y) }, for all x, y in R. That is, [A] is an interval
valued intuitionistic fuzzy subring of R. Hence, the intersection of a family of interval valued intuitionistic fuzzy
subrings of R is an interval valued intuitionistic fuzzy subring of R.
2.3 Theorem: If [A] is an interval valued intuitionistic fuzzy subring of a ring (R, +, ∙ ), then [A](x)  [A](e)
and [A](x) ≥ [A](e), for x in R, the identity element e in R.
Proof: For x in R and e is the identity element of R. Now, [A](e) = [A](xx)  rmin {[A](x), [A](x) } = [A](x).
Therefore, [A](e)  [A](x), for x in R. And, [A](e) = [A](xx)  rmax { [A](x), [A](x) } = [A](x). Therefore,
[A](e)  [A](x), for x in R.
2.4 Theorem: If [A] is an interval valued intuitionistic fuzzy subring of a ring (R, +, ∙ ), then (i) [A](xy) =
[A](e) gives [A](x) = [A](y), for x and y in R, e in R.
(ii) [A](xy) = [A](e) gives [A](x) = [A](y), for x and y in R, e in R.
Proof: Let x and y in R, the identity e in R. (i) Now, [A](x) = [A](xy+y)  rmin {[A](xy), [A](y) } = rmin {
[A](e), [A](y) } = [A](y) = [A]( x(xy) )  rmin {[A](xy), [A](x) } = rmin { [A](e), [A](x) } = [A](x).
Therefore, [A](x) = [A](y), for x, y in R. (ii) Now, [A](x) = [A](xy+y)  rmax { [A](xy), [A](y) } = rmax
{[A](e), [A](y) }= [A](y) = [A]( x(xy) )  rmax { [A](xy), [A](x) }= rmax { [A](e), [A](x) } = [A](x).
Therefore, [A](x) = [A](y), for x and y in R.

3. Interval Valued Intuitionistic Fuzzy Subrings of A Ring
2.5 Theorem: If [A] and [B] are any two interval valued intuitionistic fuzzy subrings of the rings R1 and R2
respectively, then [A]×[B] is an interval valued intuitionistic fuzzy subring of R1×R2.
Proof: Let [A] and [B] be two interval valued intuitionistic fuzzy subrings of the rings R1 and R2 respectively.
Let x1, x2 in R1 and y1, y2 in R2. Then (x1, y1) and (x2, y2) are in R1×R2. Now, [A]×[B][(x1, y1)(x2, y2)] =
[A]×[B](x1 x2, y1 y2) = rmin { [A](x1 x2), [B](y1y2)}  rmin{ rmin{ [A](x1), [A](x2) }, rmin{ [B](y1),
[B](y2)}} = rmin{ rmin {[A](x1), [B](y1) }, rmin {[A](x2), [B](y2)}} = rmin{ [A]×[B](x1, y1), [A]×[B](x2, y2)}.
Therefore, [A]×[B][(x1, y1)(x2, y2)]  rmin { [A]×[B](x1, y1), [A]×[B](x2, y2)}. Also, [A]×[B][(x1, y1)(x2, y2)] =
[A]×[B] (x1x2, y1y2) = rmin { [A](x1x2), [B](y1y2) } rmin {rmin{ [A](x1), [A](x2) }, rmin{ [B](y1), [B](y2)}} =
rmin{ rmin { [A](x1), [B](y1) }, rmin { [A](x2), [B](y2)}} = rmin { [A]×[B](x1, y1), [A]×[B](x2, y2) }. Therefore,
[A]×[B][(x1, y1)(x2, y2)]  rmin{ [A]×[B](x1, y1), [A]×[B](x2, y2)}. Now, [A]×[B][(x1, y1)(x2, y2)] =
[A]×[B](x1 x2, y1 y2) = rmax { [A](x1 x2), [B](y1 y2) } ≤ rmax{ rmax{ [A](x1), [A](x2)},
rmax{[B](y1), [B](y2)}} = rmax{ rmax{ [A](x1), [B](y1) }, rmax { [A](x2), [B](y2)}} = rmax{ [A]×[B](x1, y1),
[A]×[B](x2, y2) }. Therefore, A×B[(x1, y1)(x2, y2)] ≤ rmax{[A]×[B](x1, y1), [A]×[B](x2, y2)}. Also, [A]×[B][(x1, y1)
(x2, y2)] = [A]×[B] (x1x2, y1y2) = rmax { [A](x1x2), [B](y1y2) } ≤ rmax { rmax{ [A](x1), [A](x2) }, rmax{ [B](y1),
[B](y2)}} = rmax{ rmax { [A](x1), [B](y1)}, rmax { [A](x2), [B](y2)}} = rmax { [A]×[B](x1, y1), [A]×[B](x2, y2)}.
Therefore, [A]×[B][(x1, y1)(x2, y2)] ≤ rmax{ [A]×[B](x1, y1), [A]×[B](x2, y2)}. Hence [A]×[B] is an interval valued
intuitionistic fuzzy subring of R1×R2.
2.6 Theorem: Let [A] be an interval valued intuitionistic fuzzy subset of a ring R and [V] be the strongest
interval valued intuitionistic fuzzy relation of R. Then [A] is an interval valued intuitionistic fuzzy subring of R
if and only if [V] is an interval valued intuitionistic fuzzy subring of R×R.
Proof: Suppose that [A] is an interval valued intuitionistic fuzzy subring of a ring R. Then for any x = ( x1, x2 )
and y = ( y1, y2 ) in R×R, we have, [V](x-y) = [V] [(x1, x2) (y1, y2)] = [V](x1 y1, x2 y2) = rmin { [A](x1y1),
[A](x2y2)}  rmin { rmin {[A](x1), [A](y1) }, rmin { [A](x2), [A](y2)}} = rmin { rmin { [A](x1), [A](x2) },
rmin { [A](y1), [A](y2)}} = rmin { [V](x1, x2), [V](y1, y2) } = rmin { [V] (x), [V](y) }. Therefore, [V](xy) 
rmin { [V](x), [V](y)}, for all x, y in R×R. And, [V](xy) = [V] [(x1, x2) (y1, y2)] = [V](x1y1, x2y2) = rmin
{ [A](x1y1), [A](x2y2) }  rmin { rmin{ [A](x1), [A](y1) }, rmin{ [A](x2), [A](y2)}} = rmin{ rmin { [A](x1),
[A](x2) }, rmin{ [A](y1), [A](y2)}} = rmin { [V](x1, x2), [V](y1, y2) }= rmin{ [V](x), [V](y) }. Therefore,
[V](xy)  rmin {[V](x), [V](y) }, for all x and y in R×R. We have, [V] (xy) = [V][(x1, x2) (y1, y2)] =
[V](x1y1, x2y2) = rmax { [A](x1 y1), [A]( x2 y2) } ≤ rmax { rmax { [A](x1), [A](y1) }, rmax {[A](x2),
[A](y2)}} = rmax{ rmax { [A](x1), [A](x2) }, rmax { [A](y1), [A](y2)}} = rmax{ [V](x1, x2), [V](y1, y2)} =
rmax{[V](x), [V](y) }. Therefore,[V](xy) ≤ rmax{[V] (x), [V](y) }, for all x, y in R×R. And, [V](xy) =
[V][(x1, x2)(y1, y2)] = [V](x1y1, x2y2) = rmax { [A](x1y1), [A](x2y2) } ≤ rmax { rmax{ [A](x1), [A](y1) }, rmax
{[A](x2), [A](y2)}} = rmax{ rmax { [A](x1), [A](x2) }, rmax{ [A](y1), [A](y2) }} = rmax{ [V](x1, x2),
[V](y1, y2) }= rmax{ [V](x), [V](y) }. Therefore, [V](xy) ≤ rmax {[V](x), [V](y) }, for all x, y in R×R. This
proves that [V] is an interval valued intuitionistic fuzzy subring of R×R. Conversely assume that [V] is an
interval valued intuitionistic fuzzy subring of R×R, then for any x = (x1, x2) and y = (y1, y2) in R×R, we have
rmin { [A](x1y1), [A](x2y2) } = [V](x1 y1, x2 y2) = [V] [(x1, x2) (y1, y2)] = [V](xy)  rmin { [V] (x),
[V](y) } = rmin { [V](x1, x2), [V](y1, y2) } = rmin { rmin {[A](x1), [A](x2) }, rmin { [A](y1), [A](y2) }}. If we
put x2 = y2 = 0, we get, [A](x1y1)  rmin{[A](x1), [A](y1)}, for all x1, y1 in R. And, rmin { [A](x1y1), [A](x2y2)
} = [V](x1y1, x2y2 ) = [V][(x1, x2)(y1, y2)] = [V](xy)  rmin{ [V](x), [V](y) } = rmin {[V](x1, x2), [V](y1, y2) }
= rmin{ rmin { [A](x1), [A](x2) }, rmin { [A](y1), [A](y2)}}. If we put x2 = y2 = 0, we get [A](x1y1)  rmin{
[A](x1), [A](y1)}, for all x1, y1 in R. We have rmax{ [A](x1y1), [A](x2 y2) } = [V](x1y1, x2y2) = [V][(x1,
x2)(y1, y2)] = [V](xy) ≤ rmax{ [V](x), [V](y) }= rmax{ [V](x1, x2), [V](y1, y2)} = rmax{ rmax {[A](x1),
[A](x2) }, rmax {[A](y1), [A](y2) }}. If we put x2 = y2 = 0, we get [A](x1y1) ≤ rmax{ [A](x1), [A](y1)}, for all
x1, y1 in R. And, rmax { [A](x1y1), [A](x2y2) } = [V](x1y1, x2y2) = [V][(x1, x2)(y1, y2)] = [V](xy) ≤ rmax{[V](x),
[V](y)}= rmax{[V](x1, x2), [V](y1, y2)} = rmax{ rmax { [A](x1), [A](x2) }, rmax { [A](y1), [A](y2) }}. If we put
x2 = y2 = 0, we get [A](x1y1) ≤ rmax{ [A](x1), [A](y1) }, for all x1, y1 in R. Therefore [A] is an interval valued
intuitionistic fuzzy subring of R.
2.7 Theorem: If [A] is an interval valued intuitionistic fuzzy subring of a ring
(R, +, ∙ ), then H = { x / xR: [A](x) = [1], [A](x) = [0]} is either empty or is a subring of R.
Proof: If no element satisfies this condition, then H is empty. If x, y in H, then [A](xy)  rmin { [A](x), [A](y)
} = rmin { [1] , [1] } = [1]. Therefore, [A](xy) = [1]. And [A](xy)  rmin { [A](x), [A](y) }= rmin { [1] , [1]
} = [1]. Therefore, [A](xy) = [1]. Now, [A](xy) ≤ rmax { [A](x), [A](y) } = rmax { [0], [0] } = [0]. Therefore,
[A](xy) = [0]. And [A](xy) ≤ rmax { [A](x), [A](y) }= rmax { [0], [0] } = [0]. Therefore, [A](xy) = [0]. We
get xy, xy in H. Therefore, H is a subring of R. Hence H is either empty or is a subring of R.
2.8 Theorem: Let [A] be an interval valued intuitionistic fuzzy subring of a ring
63

4. Interval Valued Intuitionistic Fuzzy Subrings of A Ring
( R, +, ∙ ). (i) If [A](xy) = [0], then either [A](x) = [0] or [A](y) = [0], for all x, y in R. (ii) If [A](xy) = [1],
then either [A](x) = [1] or [A](y) = [1], for all x, y in R. (iii) If [A](xy) = [0], then either [A](x) = [0] or [A](y)
= [0], for all x, y in R. (iv) If [A](xy) = [1], then either [A](x) = [1] or [A](y) = [1], for all x, y in R.
Proof: Let x and y in R. (i) By the definition [A](xy)  rmin { [A](x), [A](y) }, which implies that [0]  rmin
{ [A](x), [A](y) }. Therefore, either [A](x) = [0] or [A](y) = [0]. (ii) By the definition [A](xy) ≤ rmax {
[A](x), [A](y) }, which implies that [1] ≤ rmax {[A](x), [A](y) }. Therefore, either [A](x) = [1] or [A](y) = [1].
(iii) By the definition [A](xy)  rmin { [A](x), [A](y) } which implies that [0]  rmin { [A](x), [A](y) }.
Therefore, either [A](x) = [0] or [A](y) = [0]. (iv) By the definition [A](xy) ≤ rmax {[A](x), [A](y) } which
implies that [1] ≤ rmax { [A](x), [A](y) }. Therefore, either [A](x) = [1] or [A](y) = [1].
2.9 Theorem: If [A] is an interval valued intuitionistic fuzzy subring of a ring
( R, +, ∙ ), then [A] is an interval valued intuitionistic fuzzy subring of R.
Proof: Let [A] be an interval valued intuitionistic fuzzy subring of a ring R. Consider [A] = {  x, [A](x), [A](x)
 }, for all x in R, we take [A] = [B] ={  x, [B](x), [B](x)  }, where [B](x) = [A](x), [B](x) = [1] [A](x).
Clearly, [B](xy) ≥ rmin {[B](x), [B](y) }, for all x, y in R and [B](xy) ≥ rmin { [B](x), [B](y) }, for all x, y in
R. Since [A] is an interval valued intuitionistic fuzzy subring of R, we have [A]( xy) ≥ rmin { [A](x), [A](y) },
for all x, y in R, which implies that [1]– [B](xy) ≥ rmin { [1]– [B](x), [1]– [B](y) } which implies that
[B](xy) ≤ [1]– rmin { [1]– [B](x), [1]– [B](y) } = rmax {[B](x), [B](y) }. Therefore, [B](xy) ≤ rmax {
[B](x), [B](y) }, for all x, y in R. And [A](xy) ≥ rmin { [A](x), [A](y) }, for all x, y in R, which implies that
[1]– [B](xy) ≥ rmin { [1]– [B](x), [1]– [B](y) } which implies that [B](xy) ≤ [1]– rmin { [1]– [B](x), [1]–
[B](y) } = rmax { [B](x), [B](y) }. Therefore, [B](xy) ≤ rmax { [B](x), [B](y) }, for all x, y in R. Hence [B] =
[A] is an interval valued intuitionistic fuzzy subring of R.
2.10 Theorem: If [A] is an interval valued intuitionistic fuzzy subring of a ring
( R, +, ∙ ), then [A] is an interval valued intuitionistic fuzzy subring of R.
Proof: Let [A] be an interval valued intuitionistic fuzzy subring of R. That is [A] = {  x, [A](x), [A](x)  }, for
all x in R. Let [A] = [B] = { x, [B](x), [B](x)  }, where [B](x) = [1][A](x), [B](x) = [A](x). Clearly,
[B](xy) ≤ rmax{ [B](x), [B](y) }, for all x, y in R and [B](xy) ≤ rmax{ [B](x), [B](y) }, for all x, y in R. Since
[A] is an interval valued intuitionistic fuzzy subring of R, we have [A](xy) ≤ rmax{ [A](x), [A](y) }, for all x,
y in R, which implies that [1]–[B](xy) ≤ rmax { [1]– [B](x), [1]– [B](y) } which implies that [B](xy) ≥ [1]–
rmax { [1]– [B](x), [1]– [B](y)} = rmin { [B](x), [B](y) }. Therefore, [B](xy) ≥ rmin{ [B](x), [B](y) }, for all
x, y in R. And [A](xy) ≤ rmax {[A](x), [A](y) }, for all x, y in R, which implies that [1]–[B](xy) ≤ rmax { [1]–
[B](x), [1]–[B](y) } which implies that [B](xy) ≥ [1]– rmax { [1]–[B](x), [1]–[B](y) } = rmin {[B](x), [B](y)
}. Therefore, [B](xy) ≥ rmin { [B](x), [B](y) }, for all x, y in R. Hence [B] = [A] is an interval valued
intuitionistic fuzzy subring of R.
2.11 Theorem: Let [A] be an interval valued intuitionistic fuzzy subring of a ring ( R, +, . ), then the
pseudo interval valued intuitionistic fuzzy coset (a[A])p is an interval valued intuitionistic fuzzy subring of R,
for every a in R.
Proof: Let [A] be an interval valued intuitionistic fuzzy subring of R. For every x, y in R, we have,
((a[A])p)(xy) = p(a)[A](xy) ≥ p(a) rmin {[A](x), [A](y) } = rmin {p(a)[A](x), p(a)[A](y) }= rmin { ( (a[A])p
)(x), ( (a[A])p )(y) }. Therefore, ((a[A])p ) (xy) ≥ rmin { ((a[A])p)(x), ((a[A])p)(y) }, for all x, y in R. Now,
((a[A])p)(xy) = p(a)[A](xy) ≥ p(a) rmin {[A](x), [A](y) }= rmin { p(a)[A](x), p(a)[A](y) } = rmin {((a[A])p
)(x), ( (a[A])p )(y) }. Therefore, ((a[A])p )(xy) ≥ rmin { ( (a[A])p )(x), ( (a[A])p ) (y) }, for all x, y in R. For every
x, y in R, we have, ((a[A])p )(xy) = p(a) [A](xy) ≤ p(a) rmax { [A](x), [A](y) } = rmax { p(a)[A](x),
p(a)[A](y) } = rmax {((a[A])p)(x), ((a[A])p )(y) }. Therefore, ((a[A])p)(xy) ≤ rmax { ((a[A])p )(x), ((a[A])p )(y)
}, for all x, y in R. Now, ((a[A])p )(xy) = p(a)[A](xy) ≤ p(a) rmax { [A](x), [A](y) } = rmax { p(a) [A](x), p(a)
[A](y) } = rmax { ( (a[A])p )(x), ( (a[A])p )(y) }. Therefore, ( (a[A])p )(xy) ≤ rmax { ( (a[A])p )(x), ((a[A])p)(y) },
for all x, y in R. Hence (a[A])p is an interval valued intuitionistic fuzzy subring of R.
2.12 Theorem: If [A] is an interval valued intuitionistic fuzzy subring of a ring R, then ?([A]) is an interval
valued intuitionistic fuzzy subring of R.
Proof: For every x, y in R, we have μ?([A]) (xy) = rmin { [½, ½ ], μ[A](xy) }  rmin { [½,
½], rmin { μ[A](x), μ[A](y)}}= rmin { rmin {[½, ½ ], μ[A](x) }, rmin {[½, ½ ], μ[A](y) }} = rmin{μ?( [A])(x), μ?(
[A])(y) }. Therefore μ?([A])(xy)  rmin{μ?( [A]) (x), μ?( [A]) (y) }, for all x, y in R. And μ?( [A])(xy) = rmin { [½, ½ ],
μ[A](xy)} rmin { [½, ½ ], rmin {μ[A](x), μ[A](y) } }= rmin { rmin { [½, ½ ], μ[A](x) }, rmin { [½, ½ ], μ[A](y)}} =
rmin {μ?( [A]) (x), μ?( [A]) (y) }. Therefore μ?( [A])(xy)  rmin { μ?( [A]) (x), μ?( [A]) (y) }, for all x, y in R. Also ?([A])
(xy) = rmax { [½, ½ ], [A](xy) }  rmax { [½, ½ ], rmax { [A](x), [A](y) } } = rmax { rmax {[½, ½ ], [A](x)
}, rmax {[½, ½ ], [A](y)}} = rmax { ?([A]) (x), ?([A]) (y) }. Therefore ?([A])(xy)  rmax { ?([A])(x), ?([A])(y) },
for all x, y in R. And ?([A]) (xy) = rmax { [½, ½ ], [A](xy) } rmax { [½, ½ ], rmax { [A](x), [A](y) }}
64

5. Interval Valued Intuitionistic Fuzzy Subrings of A Ring
= rmax { rmax { [½, ½ ], [A](x) }, rmax { [½, ½ ], [A](y) }} = rmax { ?([A]) (x), ?([A]) (y) }. Therefore
?([A])(xy)  rmax { ?([A])(x), ?([A])(y) }, for all x, y in R. Hence ?([A]) is an interval valued intuitionistic fuzzy
subring of R.
2.13 Theorem: If [A] is an interval valued intuitionistic fuzzy subring of a ring R, then ([A]) is an interval
valued intuitionistic fuzzy subring of R.
Proof: For every x, y in R, we have μ([A]) (xy) = rmax { [½, ½ ], μ[A](xy) } rmax {[½, ½ ], rmin { μ[A](x),
μ[A](y) } }= rmin { rmax { [½, ½ ], μ[A](x) }, rmax { [½, ½ ], μ[A](y) } } = rmin { μ([A]) (x), μ([A])(y) }. Therefore
μ([A])(xy)  rmin { μ([A])(x), μ([A])(y) }, for all x, y in R. And μ([A])(xy) = rmax { [½, ½ ], μ[A](xy) }  rmax {
[½, ½ ], rmin { μ[A](x), μ[A](y) } } = rmin { rmax { [½, ½ ], μ[A](x) }, rmax { [½, ½ ], μ[A](y) }}
= rmin { μ([A])(x), μ([A])(y) }. Therefore μ([A])(xy)  rmin { μ([A])(x), μ([A])(y) }, for all x, y in R. Also ([A])
(xy) = rmin { [½, ½ ], [A](xy) }  rmin { [½, ½ ], rmax { [A](x), [A](y) } } = rmax { rmin { [½, ½ ], [A](x)
}, rmin { [½, ½ ], [A](y) } } = rmax {([A])(x), ([A])(y) }. Therefore ([A])(xy)  rmax { ([A])(x), ([A])(y) },
for all x, y in R. And ([A]) (xy) = rmin { [½, ½ ], [A](xy) }  rmin { [½, ½ ], rmax { [A](x), [A](y) }}
= rmax { rmin{[½, ½], [A](x) }, rmin { [½, ½ ], [A](y) }} = rmax { ([A]) (x), ([A]) (y) }.
Therefore ([A])(xy)  rmax { ([A])(x), ([A])(y) }, for all x, y in R. Hence ([A]) is an interval valued
intuitionistic fuzzy subring of R.
2.14 Theorem: If [A] is an interval valued intuitionistic fuzzy subring of a ring R, then Qα ,β ([A]) is an interval
valued intuitionistic fuzzy subring of R.
Proof: For every x, y in R, for α, β D[0,1] and α +β ≤ [1], we have ( ) Q , A
 (xy) = rmin { α, μ[A](xy) }  rmin { α, rmin { μ[A](x), μ[A](y) } } = rmin { rmin { α,
 (x), ( ) Q , A
 (xy) = rmax { β, [A](xy) }  rmax { β, rmax { [A](x), [A](y) } } = rmax { rmax { β, [A](x) },
 (x), ( ) Q , A
 (y) }, for all x, y in R. Hence Qα ,β ([A]) is an interval valued intuitionistic fuzzy subring of R.
2.15 Theorem: If [A] is an interval valued intuitionistic fuzzy subring of a ring R, then Pα ,β ([A]) is an interval
valued intuitionistic fuzzy subring of R.
Proof: For every x, y in R, for α, β D[0,1] and α +β ≤ [1], we have ( ) P , A
 (xy) = rmax { α, μ[A](xy)}  rmax { α, rmin { μ[A](x), μ[A](y) } } = rmin { rmax { α, μ[A](x) }, rmax {
 (x), ( ) P , A
65
 (xy) = rmin { α, μ[A](xy) }
 (x),
rmin { α, rmin { μ[A](x), μ[A](y) }} = rmin { rmin { α, μ[A](x) }, rmin { α, μ[A](y) }} = rmin { ( ) Q , A
 Q(A(y) }. Therefore  , )
Q , ( A
)  (xy)  rmin { ( ) Q , A
 (x), ( ) Q , A
 (y) }, for all x, y in R. And
Q , (A)
μ[A](x) }, rmin { α, μ[A](y) }} = rmin { ( ) Q , A
 (y) }. Therefore ( ) Q , A
 (xy)
 (x), ( ) Q , A
 rmin { ( ) Q , A
 (y) }, for all x, y in R. Also ( ) Q , A
 (xy) = rmax { β, [A](xy) }
 rmax { β, rmax { [A](x), [A](y) }} = rmax { rmax { β, [A](x) }, rmax { β, [A](y) }} = rmax
{   (x),   (y) }. Therefore   (xy)  rmax {   (x),   (y) }, for all x, y in
Q , (A
) Q , (A
) Q , (A
) Q , (A
) Q , (A
) R. And ( ) Q , A
rmax { β, [A](y) }} = rmax { ( ) Q , A
 (y) }. Therefore ( ) Q , A
 (xy)  rmax { ( ) Q , A
 (x),
Q , (A)
 (xy) = rmax { α, μ[A](xy) } 
 (x),
rmax { α, rmin { μ[A](x), μ[A](y) }} = rmin { rmax { α, μ[A](x) }, rmax {α, μ[A](y) } } = rmin { ( ) P , A
 P(A(y) }. Therefore  , )
P , ( A
)  (xy)  rmin { ( ) P , A
 (x), ( ) P , A
 (y) }, for all x, y in R. And
P , (A)
α, μ[A](y) }} = rmin { ( ) P , A
 (y) }. Therefore ( ) P , A
 (xy)  rmin { ( ) P , A
 (x),
 (y) }, for all x, y in R. Also P(A , )
P , ( A
)  (xy) = rmin { β, [A](xy) }  rmin { β, rmax { [A](x), [A](y) }
 (x), ( ) P , A
} = rmax { rmin { β, [A](x) }, rmin { β, [A](y) }} = rmax { ( ) P , A
 (y) }. Therefore
 (xy)  rmax { P(A , )
P , ( A
)  (x), ( ) P , A
 (y) }, for all x, y in R. And ( ) P , A
 (xy)= rmin {β, [A](xy)} 
rmin { β, rmax {[A](x), [A](y)}} = rmax { rmin { β, [A](x) }, rmin { β, [A](y) } } = rmax {
 (x), P , (A)
P , ( A
)  (y) }. Therefore ( ) P , A
 (xy) rmax{ ( ) P , A
 (x), ( ) P , A
 (y)}, for all x, y in R.
Hence Pα ,β ([A]) is an interval valued intuitionistic fuzzy subring of R.
2.16 Theorem: If [A] is an interval valued intuitionistic fuzzy subring of a ring R, then Gα ,β ([A]) is an interval
valued intuitionistic fuzzy subring of R.

6. Interval Valued Intuitionistic Fuzzy Subrings of A Ring
Proof: For every x, y in R, for α, β D[0,1] and α +β ≤ [1], we have ( ) G , A
 (x), ( ) G , A
66
 (xy) = α μ[A] (xy)  α ( rmin {
μ[A](x), μ[A](y) }) = rmin { α μ[A](x), α μ[A](y)}= rmin { ( ) G , A
 (y) }. Therefore ( ) G , A
 (xy)
 (x), ( ) G , A
 rmin { ( ) G , A
 (y) }, for all x, y in R. And ( ) G , A
 (xy) = α μ[A](xy)  α ( rmin { μ[A](x),
 (x), ( ) G , A
μ[A](y) }) = rmin { α μ[A](x), α μ[A](y) } = rmin { ( ) G , A
 (y) }. Therefore
 G(A(xy)  rmin {  , )
G , ( A
)  (x), ( ) G , A
 (y) }, for all x, y in R. Also ( ) G , A
 (x-y) = β [A](xy)  β (
 (x), ( ) G , A
rmax { [A](x), [A](y) } ) = rmax { β [A](x), β [A](y) } = rmax { ( ) G , A
 (y) }. Therefore
 (x-y)  rmax { G(A , )
G , ( A
)  (x), ( ) G , A
 (y) }, for all x, y in R. And ( ) G , A
 (xy) = β [A](xy)  β
 (x), ( ) G , A
rmax { [A](x), [A](y) } ) = rmax { β [A](x), β [A](y) }= rmax { ( ) G , A
 (y) }.
 (xy)  rmax { ( ) G , A
Therefore ( ) G , A
 (x), ( ) G , A
 (y) }, for all x, y in R. Hence Gα ,β ([A]) is an interval
valued intuitionistic fuzzy subring of R.
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