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ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N.44–2020 (745–763) 745
Fuzzy bi-ideals in LA-rings
Nasreen Kausar ∗
Department of Mathematics and Statistics
University of Agriculture
Faisalabad
Pakistan
kausar.nasreen57@gmail.com
Mohammad Munir
Department of Mathematics
Government Postgraduate College
Abbottabad
Pakistan
dr.mohammadmunir@gpgc-atd.edu.pk
Badar ul Islam
Department of Electrical Engineering
NFC IEFR FSD
Pakistan
badar.utp@gmail.com
Meshari Alesemi
Department of Mathematics
Jazan University
Jazan
Kingdom of Saudi Arabia
malesemi@jazanu.edu.sa
Salahuddin
Department of Mathematics
Jazan University
Jazan
Kingdom of Saudi Arabia
drsalah12@hotmail.com
Muhammad Gulzar
Department of Mathematics
Government College University Faisalabad
Pakistan
98kohly@gmail.com
Abstract. In this paper, we give the characterizations of different classes of LA-ring
in terms of fuzzy left (resp. right, bi-, generalized bi-, (1,2)-) ideals.
Keywords: Fuzzy left (resp. right, bi-, generalized bi-, (1,2)-) ideals.
∗. Corresponding author
746 N. KAUSAR, M. MUNIR, B. UL ISLAM, M. ALESEMI, SALAHUDDIN and M. GULZAR
1. Introduction
In 1972,a generalization of abelian semigroups initiated by Naseerdin et al [6].
In ternary commutative (abelian) law: abc =cba, they introduced the braces
on the left side of this law and explored a new pseudo associative law, that
is (ab)c= (cb)a. This law (ab)c= (cb)ais called the left invertive law. A
groupoid Sis said to be a left almost semigroup (abbreviated as LA-semigroup)
if it satisfies the left invertive law : (ab)c= (cb)a. An LA-semigroup is a midway
structure between an abelian semigroup and a groupoid. Ideals in LA-semigroup
have been investigated by [14].
In [4] (resp. [1]),a groupoid Sis said to be medial (resp. paramedial) if
(ab)(cd)=(ac)(bd) (resp. (ab)(cd)=(db)(ca)).In [6], an LA-semigroup is
medial, but in general an LA-semigroup needs not to be paramedial. Every
LA-semigroup with left identity is paramedial by Stevanovic et al [14] and also
satisfies a(bc) = b(ac),(ab)(cd) = (dc)(ba).
Kamran [5], extended the notion of LA-semigroup to the left almost group
(LA-group).An LA-semigroup Sis said to be a left almost group, if there exists
left identity e∈Ssuch that ea =afor all a∈Sand for every a∈Sthere exists
b∈Ssuch that ba =e.
Rehman et al [16], initiated the concept of left almost ring (abbreviated as
LA-ring) of finitely nonzero functions which is a generalization of a commutative
semigroup ring. By a left almost ring, we mean a non-empty set Rwith at least
two elements such that (R, +) is an LA-group, (R, ·) is an LA-semigroup, both
left and right distributive laws hold. For example, from a commutative ring
(R, +,·),we can always obtain an LA-ring (R, ⊕,·) by defining for all a, b ∈R,
a⊕b=b−aand a·bis same as in the ring. Despite the fact that the structure
is non-associative and non-commutative, however it possesses properties which
usually come across in associative and commutative algebraic structures.
A non-empty subset Aof an LA-ring Ris called an LA-subring of Rif a−b
and ab ∈Afor all a, b ∈A. A is called a left (resp. right) ideal of Rif (A, +)
is an LA-group and RA ⊆A(resp. AR ⊆A). A is called an ideal of Rif it is
both a left ideal and a right ideal of R.
An LA-subring Aof Ris called a bi-ideal of Rif (AR)A⊆A. A non-empty
subset Aof Ris called a generalized bi-ideal of Rif (A, +) is an LA-group
and (AR)A⊆A. Every bi-ideal of Ris a generalized bi-ideal of R. An LA-
subring Aof Ris called a (1,2)-ideal of Rif (AR)A2⊆A.
We will initiate the concept of regular (resp. left regular, right regular, (2,2)-
regular, left weakly regular, right weakly regular, intra-regular) LA-rings. We
will also define the concept of fuzzy left (resp. right, bi-,generalized bi-, (1,2)-)
ideals.
We will describe a study of regular (resp. left regular, right regular, (2,2)-
regular, left weakly regular, right weakly regular, intra-regular) LA-rings by the
properties of fuzzy left (right, bi-, generalized bi-) ideals. In this regard, we will
prove that in regular (resp. left weakly regular) LA-rings, the concept of fuzzy
FUZZY BI-IDEALS IN LA-RINGS 747
(right, two-sided) ideals coincides. We will also show that in right regular (resp.
(2,2)-regular, right weakly regular, intra-regular) LA-rings, the concept of fuzzy
(left, right, two-sided) ideals coincides. Also in left regular LA-rings with left
identity, the concept of fuzzy (left, right, two-sided) ideals coincides. We will
also characterize left weakly regular LA-rings in terms of fuzzy right (two-sided,
bi-, generalize bi-) ideals.
2. Basic definitions and preliminary results
First time concept of fuzzy set introduced by Zadeh in his classical paper [19].
This concept has provided a useful mathematical tool for describing the behavior
of systems that are too complex to admit precise mathematical analysis by
classical methods and tools. Extensive applications of fuzzy set theory have
been found in various fields such as artificial intelligence, computer science,
management science, expert systems, finite state machines, Languages, robotics,
coding theory and others.
It soon invoked a natural question concerning a possible connection between
fuzzy sets and algebraic systems like (set, group, semigroup, ring, near-ring,
semiring, measure) theory, groupoids, real analysis, topology, differential equa-
tions and so forth. Rosenfeld [?], was the first, who introduced the concept of
fuzzy set in a group. The study of fuzzy set in semigroups was established by
Kuroki [9]. He studied fuzzy ideals and fuzzy (interior, quasi-, bi-, generalized
bi-, semiprime) ideals of semigroups. Liu [11], introduced the concept of fuzzy
subrings and fuzzy ideals of a ring. Many authors have explored the theory of
fuzzy rings (for example [3, 10, 12, 13, 18]).Gupta et al [3], gave the idea of
intrinsic product of fuzzy subsets of a ring. Kuroki [10], characterized regular
(intra-regular, both regular and intra-regular) rings in terms of fuzzy left (resp.
right, quasi, bi-) ideals.
By a fuzzy subset µof an LA-ring R, we mean a function µ:R→[0,1] and
the complement of µis denoted by µ′,is also a fuzzy subset of Rdefined by
µ′(x) = 1 −µ(x) for all x∈R.
A fuzzy subset µof an LA-ring Ris a fuzzy LA-subring of R, if µ(x−y)≥
min{µ(x), µ(y)}and µ(xy)≥min{µ(x), µ(y)}for all x, y ∈R.
Equivalent definition: A fuzzy subset µof an LA-ring Ris a fuzzy LA-
subring of R, if µ(x+y)≥min{µ(x), µ(y)}, µ(−x)≥µ(x) and µ(xy)≥
min{µ(x), µ(y)}for all x, y ∈R.
µis a fuzzy left (resp. right) ideal of R, if µ(x−y)≥min{µ(x), µ(y)}and
µ(xy)≥µ(y) (resp. µ(xy)≥µ(x)) for all x, y ∈R.
µis a fuzzy ideal of R, if it is both a fuzzy left and a fuzzy right ideal of R.
Every fuzzy ideal (whether left, right, two-sided) of Ris a fuzzy LA-subring of
R,here we are giving the example which show that the converse is not true.
748 N. KAUSAR, M. MUNIR, B. UL ISLAM, M. ALESEMI, SALAHUDDIN and M. GULZAR
Example 1. Let R={0,1,2,3,4,5,6,7}.Define + and ·in Ras follows :
+01234567
0 01234567
1 20316475
2 13025746
3 32107654
4 45670123
5 64752031
6 57461302
7 76543210
and
·01234567
000000000
104400440
204400440
300000000
403300330
507700770
607700770
703300330
Then Ris an LA-ring and µbe a fuzzy subset of R. We define µ(0) =
µ(4) = 0.7, µ(1) = µ(2) = µ(3) = µ(5) = µ(6) = µ(7) = 0.Then µis a fuzzy
LA-subring of R, but not fuzzy right ideal of Ras
µ(41) = µA(3) = 0.
µ(4) = 0.7.
⇒µ(41) µ(4).
A fuzzy LA-subring µof an LA-ring Ris a fuzzy bi-ideal of Rif µ((xa)y)≥
min{µ(x), µ(y)}for all x, y, a ∈R.
A fuzzy subset µof an LA-ring Ris a fuzzy generalized bi-ideal of Rif µ(x−
y)≥min{µ(x), µ(y)}and µ((xa)y)≥min{µ(x), µ(y)}for all x, y, a ∈R. Every
fuzzy bi-ideal of Ris a fuzzy generalized bi-ideal of R.
A fuzzy LA-subring µof an LA-ring Ris called a fuzzy (1,2)-ideal of Rif
µ((xa)(yz)) ≥min {µ(x), µ(y), µ(z)}for all x, y, z, a ∈R.
Let Abe a non-empty subset of an LA-ring R. Then the characteristic func-
tion of Ais denoted by χAand defined as
χA(x) = 1,if x∈A
0,if x /∈A
The product of two fuzzy subsets µand νis denoted by µ◦νand defined
as:
(µ◦ν)(x) =
x=
n
∑
i=1
aibi
{
n
i=1
{µ(ai)∧ν(bi)}},if x=
n
i=1
aibi, ai, bi∈R
0,if x̸=
n
i=1
aibi.
Now we are giving the some fundamental properties, which will be very
helpful in the next section.
FUZZY BI-IDEALS IN LA-RINGS 749
Theorem 1. Let Aand Bbe two non-empty subsets of an LA-ring R. Then
the following conditions hold.
(1) If A⊆Bthen χA⊆χB.
(2) χA◦χB=χAB.
(3) χA∪χB=χA∪B.
(4) χA∩χB=χA∩B.
Proof. Straight forward.
Example 2. Let R={a, b, c, d}.Define + and ·in Ras follows :
+a b c d
a a b c d
b d a b c
c c d a b
d b c d a
and
·a b c d
aaaaa
b a b a b
c a a c c
d a b c d
Then Ris an LA-ring and µbe a fuzzy subset of R. We define µ(a) = µ(c) =
0.7, µ(b) = µ(d) = 0.Then µis a fuzzy ideal of R.
Lemma 2.1. Every fuzzy left (resp. right, two-sided)ideal of an LA-ring Ris
a fuzzy bi-ideal of R. But the converse is not true in general.
Proof. Straight forward.
Example 3. Above µin example 1 is also a fuzzy bi-ideal of R,but not right
ideal of R.
Lemma 2.2. Every fuzzy bi-ideal of an LA-ring Ris a fuzzy (1,2)-ideal of R.
Proof. Straight forward.
Remark 1. Every fuzzy left (resp. right, two-sided) ideal of an LA-ring Ris a
fuzzy (1,2)-ideal of R.
Proposition 2.1. Let Rbe an LA-ring having the property a=a2for every
a∈R. Then every fuzzy (1,2)-ideal of Ris a fuzzy bi-ideal of R.
Proof. Suppose that µis a fuzzy (1,2)-ideal of Rand a, x, y ∈R. Thus
µ((xa)y) = µ((xa)(yy)) ≥min{µ(x), µ(y), µ(y)}= min{µ(x), µ(y)}.
Therefore µis a fuzzy bi-ideal of R.
Lemma 2.3. Let Rbe an LA-ring and ∅ ̸=A⊆R. Then Ais an LA-subring
of Rif and only if the characteristic function χAof Ais a fuzzy LA-subring of
R.
Proof. Straight forward.
750 N. KAUSAR, M. MUNIR, B. UL ISLAM, M. ALESEMI, SALAHUDDIN and M. GULZAR
Lemma 2.4. Let Rbe an LA-ring and ∅ ̸=A⊆R. Then Ais a left (resp.
right)ideal of Rif and only if the characteristic function χAof Ais a fuzzy left
(resp. right)ideal of R.
Proof. Straight forward.
Proposition 2.2. Let Rbe an LA-ring and ∅ ̸=A⊆R. Then Ais a (1,2)-ideal
of Rif and only if the characteristic function χAof Ais a fuzzy (1,2)-ideal of
R.
Proof. Let Abe a (1,2)-ideal of R, this implies that Ais an LA-subring of
R. Then χAis a fuzzy LA-subring of R, by the Lemma 2.3. Let a, x, y, z ∈
R. If x, y, z ∈A, then by definition χA(x) = 1 = χA(y) = χA(z).Since
(xa)(yz)∈A, A being a (1,2)-ideal, so χA((xa)(yz)) = 1.Thus χA((xa)(yz)) ≥
min{χA(x), χA(y), χA(z)}.Similarly, we have χA((xa)(yz)) ≥min{χA(x), χA(y),
χA(z)},when x, y, z /∈A. Hence the characteristic function χAof Ais a fuzzy
(1,2)-ideal of R.
Conversely, assume that the characteristic function χAof Ais a fuzzy (1,2)-
ideal of R, this implies that χAis a fuzzy LA-subring of R. Then Ais an LA-
subring of Rby the Lemma 2.3. Let t∈(AR)A2,so t= (xa)(yz),where
x, y, z ∈A, a ∈R. So by definition χA(x) = 1 = χA(y) = χA(z).Since
χA((xa)(yz)) ≥χA(x)∧χA(y)∧χA(z)=1, χAbeing a fuzzy (1,2)-ideal of
R. Thus χA((xa)(yz)) = 1,i.e., (xa)(yz)∈A. Hence Ais a (1,2)-ideal of R.
Remark 2. Let Rbe an LA-ring and ∅ ̸=A⊆R. Then Ais a bi-ideal of Rif
and only if the characteristic function χAof Ais a fuzzy bi-ideal of R.
Zadeh [19], introduced the concept of level set. Das [2], studied the fuzzy
groups, level subgroups and gave the proper definition of a level set such that:
let µbe a fuzzy subset of a non-empty set S, for t∈[0,1],the set µt={x∈S|
µ(x)≥t},is called a level subset of the fuzzy subset µ.
Let µbe a fuzzy subset of an LA-ring R, then for all t∈(0,1],we define a
set U(µ;t) = {x∈R|µ(x)≥t},which is called an upper t-level of µ.
Lemma 2.5. Let µbe a fuzzy subset of an LA-ring R. Then µis a fuzzy LA-
subring of Rif and only if upper t-level U(µ;t)of µis an LA-subring of Rfor
all t∈(0,1].
Proof. Straight forward.
Lemma 2.6. Let µbe a fuzzy subset of an LA-ring R. Then µis a fuzzy left
(resp. right)ideal of Rif and only if upper t-level U(µ;t)of µis a left (resp.
right)ideal of Rfor all t∈(0,1].
Proof. Straight forward.
FUZZY BI-IDEALS IN LA-RINGS 751
Proposition 2.3. Let µbe a fuzzy subset of an LA-ring R. Then µis a fuzzy
(1,2)-ideal of Rif and only if upper t-level U(µ;t)of µis a (1,2)-ideal of Rfor
all t∈(0,1].
Proof. Suppose that µis a fuzzy (1,2)-ideal of R, this implies that µis a
fuzzy LA-subring of R. Then U(µ;t) is an LA-subring of Rby the Lemma
2.5. Let x, y, z ∈U(µ;t) and a∈R, then by definition µ(x), µ(y), µ(z)≥t.
Now µ((xa)(yz)) ≥µ(x)∧µ(y)∧µ(z)≥t, µ being a fuzzy (1,2)-ideal of R, i.e.,
(xa)(yz)∈U(µ;t).Therefore U(µ;t) is a (1,2)-ideal of R.
Conversely, assume that U(µ;t) is a (1,2)-ideal of R, this means that U(µ;t)
is an LA-subring of R. Then µis a fuzzy LA-subring of Rby the Lemma 2.5.
We have to show that µ((xa)(yz)) ≥µ(x)∧µ(y)∧µ(z).We suppose a con-
tradiction µ((xa)(yz)) < µ(x)∨µ(y)∨µ(z).Let µ(x) = t=µ(y) = µ(z),so
µ(x), µ(y), µ(z)≥t, i.e., x, y, z ∈U(µ;t).But µ((xa)(yz)) < t, i.e., (xa)(yz)/∈
U(µ;t),which is a contradiction. Therefore µ((xa)(yz)) ≥µ(x)∧µ(y)∧µ(z).
Remark 3. Let µbe a fuzzy subbset of an LA-ring R. Then µis a fuzzy bi-ideal
of Rif and only if upper t-level U(µ;t) of µis a bi-ideal of Rfor all t∈(0,1].
3. Characterizations of LA-rings
In this section, we characterize different classes of LA-ring in terms of fuzzy
left (right, bi-, generalized bi-) ideals. An LA-ring Ris a regular, if for every
element x∈R, there exists an element a∈Rsuch that x= (xa)x. An LA-ring
Ris an intra-regular, if for every element x∈R, there exist elements ai, bi∈R
such that x=n
i=1(aix2)bi.
An LA-ring Ris a left (resp. right) regular, if for every element x∈R,
there exists an element a∈Rsuch that x=ax2(resp. x2a).An LA-ring Ris
completely regular if it is regular, left regular and right regular.
An LA-ring Ris a (2,2)-regular if for every element x∈R, there exists an
element a∈Rsuch that x= (x2a)x2.
An LA-ring Ris a locally associative LA-ring if (a.a).a =a.(a.a) for all
a∈R.
A ring Ris a left (resp. right) weakly regular if I2=Ifor every left (resp.
right) ideal Iof R, equivalently x∈RxRx(x∈xRxR) for every x∈R. An
LA-ring Ris called a weakly regular if it is both a left weakly regular and a
right weakly regular [15].
Now we define this notion in a class of non-associative and non-commutative
rings (LA-ring).
An LA-ring Ris called a left (resp. right) weakly regular if for every element
x∈R, there exist elements a, b ∈Rsuch that x= (ax)(bx) (resp. x= (xa)(xb)).
An LA-ring Ris called a weakly regular if it is both a left weakly regular and a
right weakly regular.
Lemma 3.1. Every fuzzy right ideal of an LA-ring Rwith left identity e, is a
fuzzy ideal of R.
752 N. KAUSAR, M. MUNIR, B. UL ISLAM, M. ALESEMI, SALAHUDDIN and M. GULZAR
Proof. Let µbe a fuzzy right ideal of Rand x, y ∈R. Thus µ(xy) = µ((ex)y) =
µ((yx)e)≥µ(yx)≥µ(y).Hence µis a fuzzy ideal of R.
Lemma 3.2. Every fuzzy right ideal of a regular LA-ring Ris a fuzzy ideal of
R.
Proof. Suppose that µis a fuzzy right ideal of R. Let x, y ∈R, this implies
that there exists a∈R, such that x= (xa)x. Thus µ(xy) = µ(((xa)x)y) =
µ((yx)(xa)) ≥µ(yx)≥µ(y).Therefore µis a fuzzy ideal of R.
Proposition 3.1. Let Rbe a regular LA-ring having the property a=a2for
every a∈R, with left identity e. Then every fuzzy generalized bi-ideal of Ris a
fuzzy bi-ideal of R.
Proof. Let µbe a fuzzy generalized bi-ideal of Rand x, y ∈R, this implies
that there exists a∈Rsuch that x= (xa)x. We have to show that µis a fuzzy
LA-subring of R. Thus
µ(xy) = µ(((xa)x)y) = µ(((xa)x2)y) = µ(((xa)(xx))y)
=µ((x((xa)x))y)≥min{µ(x), µ(y)}.
Hence µis a fuzzy LA-subring of R.
Lemma 3.3 ([7, Lemma 8]).Let Rbe an LA-ring with left identity e. Then Ra
is the smallest left ideal of Rcontaining a.
Lemma 3.4 ([7, Lemma 9]).Let Rbe an LA-ring with left identity e. Then aR
is a left ideal of R.
Proposition 3.2 ([7, Proposition 5]).Let Rbe an LA-ring with left identity e.
Then aR ∪Ra is the smallest right ideal of Rcontaining a.
Lemma 3.5. Let Rbe an LA-ring. Then µ◦ν⊆µ∩νfor every fuzzy right
ideal µand for every fuzzy left ideal νof R.
Proof. Let µbe a fuzzy right and νbe a fuzzy left ideal of Rand x∈R. If x
cannot be expressible as x=n
i=1 aibi,where ai, bi∈Rand nis any positive
integer, then obviously µ◦ν⊆µ∩ν, otherwise we have
(µ◦ν) (x) =
x=∑n
i=1 aibi
{
n
i=1
{µ(ai)∧ν(bi)}}
≤
x=∑n
i=1 aibi
{
n
i=1
{µ(aibi)∧ν(aibi)}}
=
x=∑n
i=1 aibi
{
n
i=1
(µ∩ν) (aibi)}= (µ∩ν) (x).
⇒µ◦ν⊆µ∩ν.
FUZZY BI-IDEALS IN LA-RINGS 753
Theorem 2. Let Rbe an LA-ring with left identity e, such that (xe)R=xR
for all x∈R. Then the following conditions are equivalent.
(1) Ris a regular.
(2) µ∩ν=µ◦νfor every fuzzy right ideal µand for every fuzzy left ideal
νof R.
Proof. Suppose that (1) holds. Since µ◦ν⊆µ∩ν, for every fuzzy right ideal
µand every fuzzy left ideal νof Rby the Lemma 3.5. Let x∈R, this implies
that there exists an element a∈Rsuch that x= (xa)x. Thus
(µ◦ν)(x) =
x=∑n
i=1 aibi
{∧n
i=1 {µ(ai)∧ν(bi)}}
≥min{µ(xa), ν(x)} ≥ min{µ(x), ν (x)}
= (µ∧ν)(x) = (µ∩ν)(x).
⇒µ∩ν⊆µ◦ν.
Hence µ∩ν=µ◦ν, i.e., (1) ⇒(2) .Assume that (2) is true and a∈R.
Then Ra is a left ideal of Rcontaining aby the Lemma 3.3 and aR ∪Ra
is a right ideal of Rcontaining aby the Proposition 3.2. So χRa is a fuzzy
left ideal and χaR∪Ra is a fuzzy right ideal of R, by the Lemma 2.4. By our
assumption χaR∪Ra ∩χRa =χaR∪Ra ◦χRa,i.e., χ(aR∪Ra)∩Ra =χ(aR∪Ra)Ra.Thus
(aR∪Ra)∩Ra = (aR ∪Ra)Ra. Since a∈(aR ∪Ra)∩Ra, i.e., a∈(aR∪Ra)Ra,
so a∈(aR)(Ra)∪(Ra)(Ra).Now (Ra)(Ra) = ((Re)a)(Ra) = ((ae)R)(Ra) =
(aR)(Ra).This implies that
(aR)(Ra)∪(Ra)(Ra) = (aR)(Ra)∪(aR)(Ra) = (aR)(Ra).
Thus a∈(aR)(Ra).Then
a= (ax)(ya) = ((ya)x)a= (((ey)a)x)a= (((ay)e)x)a
= ((xe)(ay))a= (a((xe)y))a∈(aR)a, for any x, y ∈R.
This means that a∈(aR)a, i.e., ais regular. Hence Ris a regular, i.e.,
(2) ⇒(1) .
Theorem 3. Let Rbe a regular locally associative LA-ring having the property
a=a2for every a∈R. Then for every fuzzy bi-ideal µof R, µ(an) = µ(a2n)for
all a∈R, where nis any positive integer.
Proof. For n= 1.Let a∈R, this implies that there exists an element x∈R
such that a= (ax)a. Now a= (ax)a= (a2x)a2,because a=a2.Thus
µ(a) = µ(a2x)a2≥min{µa2, µ a2}=µa2
=µ(aa)≥min{µ(a), µ (a)}=µ(a).
⇒µ(a) = µa2.
754 N. KAUSAR, M. MUNIR, B. UL ISLAM, M. ALESEMI, SALAHUDDIN and M. GULZAR
Now a2=aa = ((a2x)a2)((a2x)a2) = (a4x2)a4,then the result is true for
n= 2.Suppose that the result is true for n=k, i.e., µ(ak) = µa2k.Now
ak+1 =aka= ((a2kxk)a2k)((a2x)a2) = (a2(k+1)xk+1)a2(k+1).Thus
µak+1=µ(a2(k+1)xk+1)a2(k+1) ≥min{µ(a2(k+1)), µ a2(k+1)}
=µa2(k+1)=µak+1ak+1)
≥min{µak+1), µ ak+1)}=µ(ak+1)).
⇒µ(ak+1) = µ(a2(k+1) ).
Hence by induction method, the result is true for all positive integers.
Lemma 3.6. Let Rbe a (2,2)-regular LA-ring. Then every fuzzy left (resp.
right)ideal of Ris a fuzzy ideal of R.
Proof. Let µbe a fuzzy right ideal of Rand x, y ∈R, this implies that there
exists an element a∈Rsuch that x= (x2a)x2.Thus µ(xy) = µ(((x2a)x2)y) =
µ((yx2)(x2a)) ≥µ(yx2)≥µ(y).Hence µis a fuzzy ideal of R. Similarly, for left
ideal.
Remark 4. The concept of fuzzy (left, right, two-sided) ideals coincides in
(2,2)-regular LA-rings.
Proposition 3.3. Every fuzzy generalized bi-ideal of (2,2)-regular LA-ring R
with left identity e, is a fuzzy bi-ideal of R.
Proof. Suppose that µis a fuzzy generalized bi-ideal of Rand x, y ∈R, this
means that there exists an element a∈Rsuch that x= (x2a)x2.We have to
show that µis a fuzzy LA-subring of R. Thus
µ(xy) = µ(((x2a)x2)y) = µ(((x2a)(xx))y)
=µ((x((x2a)x))y)≥min{µ(x), µ(y)}.
Therefore µis a fuzzy LA-subring of R.
Theorem 4. Let Rbe a (2,2)-regular locally associative LA-ring. Then for
every fuzzy bi-ideal µof R, µ(an) = µ(a2n)for all a∈R, where nis any
positive integer.
Proof. Same as Theorem 3.
Lemma 3.7. Let Rbe a right regular LA-ring. Then every fuzzy left (resp.
right)ideal of Ris a fuzzy ideal of R.
FUZZY BI-IDEALS IN LA-RINGS 755
Proof. Let µbe a fuzzy right ideal of Rand x, y ∈R, this implies that there
exists a∈Rsuch that x=x2a. Thus
µ(xy) = µ((x2a)y) = µ(((xx)a)y) = µ(((ax)x)y)
=µ((yx)(ax)) ≥µ(yx)≥µ(y).
Hence µis a fuzzy ideal of R. Similarly, for left ideal.
Remark 5. The concept of fuzzy (left, right, two-sided) ideals coincides in right
regular LA-rings.
Proposition 3.4. Every fuzzy generalized bi-ideal of a right regular LA-ring R
with left identity e, is a fuzzy bi-ideal of R.
Proof. Suppose that µis a fuzzy generalized bi-ideal of Rand x, y ∈R, this
means that there exists a∈Rsuch that x=x2a. We have to show that µis a
fuzzy LA-subring of R. Thus
µ(xy) = µ((x2a)y) = µ(((xx)(ea))y) = µ(((ae)(xx))y)
=µ((x((ae)x))y)≥min{µ(x), µ(y)}.
Therefore µis a fuzzy LA-subring of R.
Lemma 3.8. Let Rbe a left regular LA-ring with left identity e. Then every
fuzzy left (resp. right)ideal of Ris a fuzzy ideal of R.
Proof. Assume that µis a fuzzy right ideal of Rand x, y ∈R, then there exists
an element a∈Rsuch that x=ax2.Thus
µ(xy) = µ((ax2)y) = µ((a(xx))y) = µ((x(ax))y)
=µ(y(ax))x)≥µ(y(ax)) ≥µ(y).
So µis a fuzzy ideal of R. Similarly, for left ideal.
Remark 6. The concept of fuzzy (left, right, two-sided) ideals coincides in left
regular LA-rings with left identity.
Proposition 3.5. Every fuzzy generalized bi-ideal of a left regular LA-ring R
with left identity e, is a fuzzy bi-ideal of R.
Proof. Let µbe a fuzzy generalized bi-ideal of Rand x, y ∈R, this implies
that there exists a∈Rsuch that x=ax2.We have to show that µis a fuzzy
LA-subring of R. Thus µ(xy) = µ((ax2)y) = µ((a(xx))y) = µ((x(ax))y)≥
min{µ(x), µ(y)}.Hence µis a fuzzy LA-subring of R.
Theorem 5. Let Rbe a regular and right regular locally associative LA-ring.
Then for every fuzzy right ideal µof R, µ(an) = µ(a3n)for all a∈R, where n
is any positive integer.
756 N. KAUSAR, M. MUNIR, B. UL ISLAM, M. ALESEMI, SALAHUDDIN and M. GULZAR
Proof. For n= 1.Let a∈R, this means that there exists an element x∈R
such that a= (ax)aand a=a2x. Now a= (ax)a= (ax)(a2x) = a3x2.Thus
µ(a) = µ(a3x2)≥µ(a3) = µ(aa2)≥min{µ(a), µ a2}
≥min{µ(a), µ (a), µ (a)}=µ(a).
⇒µ(a) = µa3.
Here a2=aa = (a3x2)(a3x2) = a6x4,then the result is true for n= 2.
Assume that the result is true for n=k, i.e., µ(ak) = µ(a3k).Now ak+1 =
aka= (a3kx2k)(a3x2) = a3(k+1)x2(k+1) .Thus
µ(ak+1) = µ(a3(k+1) x2(k+1))≥µ(a3(k+1) ) = µ(a3k+3)
=µ(ak+1a2k+2)≥min{µak+1, µ a2k+2 }
≥min{µak+1, µ ak+1, µ ak+1}=µak+1 .
⇒µ(ak+1) = µ(a3(k+1) ).
Hence by induction method, the result is true for all positive integers.
Theorem 6. Let Rbe a right regular locally associative LA-ring. Then for
every fuzzy right ideal µof R, µ(an) = µ(a2n)for all a∈R, where nis any
positive integer.
Proof. For n= 1.Let a∈R, then there exists an element x∈Rsuch that
a=a2x. Thus
µA(a) = µA(a2x)≥µA(a2) = µA(aa)
≥min{µA(a), µA(a)}=µA(a)⇒µ(a) = µa2.
Now a2=aa = (a2x)(a2x) = a4x2,then the result is true for n= 2.Suppose
that the result is true for n=k, i.e., µ(ak) = µ(a2k).Now ak+1 =aka=
(a2kxk)(a2x) = a2(k+1)x(k+1).Thus
µ(ak+1) = µ(a2(k+1) x(k+1))≥µ(a2(k+1) )
=µ(a2k+2) = µ(ak+1 ak+1)
≥min{µak+1, µ ak+1}=µak+1⇒µ(ak+1 ) = µ(a2(k+1)).
Hence by induction method, the result is true for all positive integers.
Lemma 3.9. Let Rbe a right regular locally associative LA-ring with left iden-
tity e. Then for every fuzzy right ideal µof R, µ(ab) = µ(ba)for all a, b ∈R.
Proof. Let a, b ∈R. By using Theorem 6 (for n= 1).Now
µ(ab) = µ((ab)2) = µ((ab)(ab))
=µ((ba)(ba)) = µ((ba)2) = µ(ba).
FUZZY BI-IDEALS IN LA-RINGS 757
Remark 7. It is easy to see that, if Ris a left regular locally associative LA-ring
with left identity e. Then for every fuzzy left ideal µof R, µ(an) = µ(a2n) for
all a∈R, where nis any positive integer. And also for every fuzzy left ideal µ
of R, µ(ab) = µ(ba) for all a, b ∈R.
Lemma 3.10. Let Rbe a right weakly regular LA-ring. Then every fuzzy left
(resp. right)ideal of Ris a fuzzy ideal of R.
Proof. Suppose that µis a fuzzy right ideal of Rand x, y ∈R, this means that
there exist a, b ∈Rsuch that x= (xa)(xb).Thus
µ(xy) = µ(((xa)(xb))y) = µ((((xb)a)x)y)
=µ((((ab)x)x)y) = µ((yx)((ab)x))
=µ((yx)(nx)) say ab =n
≥µ(yx)≥µ(y).
Therefore µis a fuzzy ideal of R. Similarly, for left ideal.
Remark 8. The concept of fuzzy (left, right, two-sided) ideals coincides in right
weakly regular LA-rings.
Proposition 3.6. Every fuzzy generalized bi-ideal of a right weakly regular LA-
ring Rwith left identity e, is a fuzzy bi-ideal of R.
Proof. Assume that µis a fuzzy generalized bi-ideal of Rand x, y ∈R, then
there exist elements a, b ∈Rsuch that x= (xa)(xb).We have to show that µ
is a fuzzy LA-subring of R. Thus µ(xy) = µ(((xa)(xb))y) = µ((x((xa)b))y)≥
min{µ(x), µ(y)}.So µis a fuzzy LA-subring of R.
Lemma 3.11. Let Rbe a left weakly regular LA-ring. Then every fuzzy right
ideal of Ris a fuzzy ideal of R.
Proof. Let µbe a fuzzy right ideal of Rand x, y ∈R, this implies that
there exist a, b ∈Rsuch that x= (ax)(bx).Thus µ(xy) = µ(((ax)(bx))y) =
µ(y(bx))(ax)≥µ(y(bx)) ≥µ(y).Hence µis a fuzzy ideal of R.
Lemma 3.12. Let Rbe a left weakly regular LA-ring with left identity e. Then
every fuzzy left ideal of Ris a fuzzy ideal of R.
Proof. Suppose that µis a fuzzy left ideal of Rand x, y ∈R, this means that
there exist a, b ∈Rsuch that x= (ax)(bx).Thus
µ(xy) = µ(((ax)(bx))y) = µ(((ab)(xx))y)
=µ((x((ab)x))y) = µ((y((ab)x))x)≥µ(x).
Therefore µis a fuzzy ideal of R.
758 N. KAUSAR, M. MUNIR, B. UL ISLAM, M. ALESEMI, SALAHUDDIN and M. GULZAR
Proposition 3.7. Every fuzzy generalized bi-ideal of a left weakly regular LA-
ring Rwith left identity e, is a fuzzy bi-ideal of R.
Proof. Assume that µis a fuzzy generalized bi-ideal of Rand x, y ∈R, then
there exist elements a, b ∈Rsuch that x= (ax)(bx).We have to show that µis
a fuzzy LA-subring of R. Thus
µ(xy) = µ(((ax)(bx))y) = µ(((ab)(xx))y)
=µ((x((ab)x))y)≥min{µ(x), µ(y)}.
So µis a fuzzy LA-subring of R.
Remark 9. It is easy to see that, if Ris a left (resp. right) weakly regular
locally associative LA-ring. Then for every fuzzy left (resp. right) ideal µof R,
µ(an) = µ(a2n) for all a∈R, where nis any positive integer.
Theorem 7. Let Rbe an LA-ring with left identity e, such that (xe)R=xR
for all x∈R. Then the following conditions are equivalent.
(1) Ris a left weakly regular.
(2) µ∩ν=µ◦νfor every fuzzy right ideal µand for every fuzzy left ideal
νof R.
Proof. Suppose that (1) holds. Since µ◦ν⊆µ∩ν, for every fuzzy right ideal
µand every fuzzy left ideal νof Rby the Lemma 3.5. Let x∈R, this implies
that there exist a, b ∈Rsuch that x= (ax)(bx) = (ab)(xx) = x((ab)x).Now
(µ◦ν)(x) =
x=∑n
i=1 aibi
{
n
i=1
{µ(ai)∧ν(bi)}}
≥µ(x)∧ν((ab)x)≥µ(x)∧ν(x) = (µ∩ν)(x).
⇒µ∩ν⊆µ◦ν.
Hence µ∩ν=µ◦ν, i.e., (1) ⇒(2) .Assume that (2) is true and a∈R.
Then Ra is a left ideal of Rcontaining aby the Lemma 3.3 and aR ∪Ra is
a right ideal of Rcontaining aby the Proposition 3.2. So χRa is a fuzzy left
ideal and χaR∪Ra is a fuzzy right ideal of R, by the Lemma 2.4. Then by our
assumption χaR∪Ra ∩χRa =χaR∪Ra ◦χRa,i.e., χ(aR∪Ra)∩Ra =χ(aR∪Ra)Ra by the
Theorem 1. Thus (aR∪Ra)∩Ra = (aR ∪Ra)Ra. Since a∈(aR ∪Ra)∩Ra, i.e.,
a∈(aR∪Ra)Ra, so a∈(aR)(Ra)∪(Ra)(Ra).This implies that a∈(aR)(Ra) or
a∈(Ra)(Ra).If a∈(Ra)(Ra),then Ris a left weakly regular. If a∈(aR)(Ra),
then
(aR)(Ra) = ((ea)(RR))(Ra) = ((RR)(ae))(Ra)
= (((ae)R)R)(Ra) = ((aR)R)(Ra)
= ((RR)a)(Ra) = (Ra)(Ra).
Hence Ris a left weakly regular, i.e., (2) ⇒(1) .
FUZZY BI-IDEALS IN LA-RINGS 759
Theorem 8. Let Rbe an LA-ring with left identity e, such that (xe)R=xR
for all x∈R. Then the following conditions are equivalent.
(1) Ris a left weakly regular.
(2) µ∩γ⊆µ◦γfor every fuzzy bi-ideal µand for every fuzzy ideal γof R.
(3) ν∩γ⊆ν◦γfor every fuzzy generalized bi-ideal νand for every fuzzy
ideal γof R.
Proof. Assume that (1) holds. Let νbe a fuzzy generalized bi-ideal and γbe
a fuzzy ideal of R. Let x∈R, this means that there exist a, b ∈Rsuch that
x= (ax)(bx) = (ab)(xx) = x((ab)x).Thus
(µ◦γ)(x) =
x=∑n
i=1 aibi
{
n
i=1
{µ(ai)∧γ(bi)}}
≥µ(x)∧γ((ab)x)
≥µ(x)∧γ(x) = (µ∩γ)(x).
⇒ν∩γ⊆ν◦γ.
Therefore (1) ⇒(3) .It is clear that (3) ⇒(2) .Suppose that (2) holds.
Then µ∩γ⊆µ◦γ, where µis a fuzzy right ideal of R. Since µ◦γ⊆µ∩γ,
so µ◦γ=µ∩γ. Therefore Ris a left weakly regular by the Theorem 7, i.e.,
(2) ⇒(1) .
Theorem 9. Let Rbe an LA-ring with left identity e, such that (xe)R=xR
for all x∈R. Then the following conditions are equivalent.
(1) Ris a left weakly regular.
(2) µ∩γ∩δ⊆(µ◦γ)◦δfor every fuzzy bi-ideal µ, every fuzzy ideal γand
for every fuzzy right ideal δof R.
(3) ν∩γ∩δ⊆(ν◦γ)◦δfor every fuzzy generalized bi-ideal ν, every fuzzy
ideal γand for every fuzzy right ideal δof R.
Proof. Suppose that (1) holds. Let νbe a fuzzy generalized bi-ideal, γbe a
fuzzy ideal and δbe a fuzzy right ideal of R. Let x∈R, then there exist elements
a, b ∈Rsuch that x= (ax)(bx).Now
x= (ax)(bx) = (xb)(xa)
xb = ((ax)(bx))b= ((xx)(ba))b
= (b(ba))(xx) = c(xx) = x(cx) say c=b(ba)
Thus
((ν◦γ)◦δ)(x) =
x=∑n
i=1 aibi
{
n
i=1
{(ν◦γ) (ai)∧δ(bi)}}
≥(ν◦γ)(xb)∧δ(xa)
760 N. KAUSAR, M. MUNIR, B. UL ISLAM, M. ALESEMI, SALAHUDDIN and M. GULZAR
≥(ν◦γ)(xb)∧δ(x)
=
xb=∑n
i=1 piqi
{
n
i=1
{ν(pi)∧γ(qi)}} ∧ δ(x)
≥ν(x)∧γ(cx)∧δ(x)
≥ν(x)∧γ(x)∧δ(x) = (µ∩γ∩δ)(x).
⇒µ∩γ∩δ⊆(ν◦γ)◦δ.
Hence (1) ⇒(3) .Since (3) ⇒(2) ,every fuzzy bi-ideal of Ris a fuzzy
generalized bi-ideal of R. Assume that (2) is true. Then µ∩γ∩R⊆(µ◦γ)◦R,
where µis a fuzzy right ideal of R, i.e., µ∩γ⊆µ◦γ. Since µ◦γ⊆µ∩γ,
so µ◦γ=µ∩γ. Hence Ris a left weakly regular by the Theorem 7, i.e.,
(2) ⇒(1) .
Lemma 3.13. Let Rbe an intra-regular LA-ring. Then every fuzzy left (resp.
right)ideal of Ris a fuzzy ideal of R.
Proof. Suppose that µis a fuzzy right ideal of R. Let x, y ∈R, this im-
plies that there exist ai, bi∈R, such that x=n
i=1(aix2)bi.Thus µ(xy) =
µ(((aix2)bi)y) = µ((ybi)(aix2)) ≥µ(ybi)≥µ(y).Hence µis a fuzzy ideal of R.
Similarly, for left ideal.
Proposition 3.8. Every fuzzy generalized bi-ideal of an intra-regular LA-ring
Rwith left identity e, is a fuzzy bi-ideal of R.
Proof. Let µbe a fuzzy generalized bi-ideal of Rand x, y ∈R, this implies
that there exist ai, bi∈Rsuch that x=n
i=1(aix2)bi.We have to show that µ
is a fuzzy LA-subring of R. Now
x= (aix2)bi= (aix2)(ebi) = (aie)(x2bi)
= (aie)((xx)bi) = (aie)((bix)x) = (x(bix))(eai)
= (x(bix))ai= (ai(bix))x= (ai(bix))(ex)
= (xe)((bix)ai) = (bix)((xe)ai) = (bix)((aie)x)
= (x(aie))(xbi) = x((x(aie))bi) = xn, say n= (x(aie))bi
Thus µ(xy) = µ((xn)y)≥min{µ(x), µ(y)}.Hence µis a fuzzy LA-subring
of R.
Theorem 10. Let Rbe an LA-ring with left identity e, such that (xe)R=xR
for all x∈R. Then the following conditions are equivalent.
(1) Ris an intra-regular.
(2) µ∩ν⊆µ◦νfor every fuzzy right ideal νand for every fuzzy left ideal
µof R.
FUZZY BI-IDEALS IN LA-RINGS 761
Proof. Assume that (1) holds. Let x∈R, then there exist elements ai, bi∈R
such that x=n
i=1(aix2)bi.Now
x= (aix2)bi= (ai(xx))bi= (x(aix))(ebi)
= (xe)((aix)bi) = (aix)((xe)bi).
Thus
(µ◦ν)(x) =
x=∑n
i=1 aibi
{
n
i=1
µ(ai)∧ν(bi)}
≥min{µ(aix), ν((xe)bi)} ≥ min{µ(x), ν (x)}
= (µ∧ν)(x) = (µ∩ν)(x).
⇒µ∩ν⊆µ◦ν.
Hence (1) ⇒(2) .Suppose that (2) is true and a∈R. Then Ra is a left
ideal of Rcontaining aby the Lemma 3.3 and aR ∪Ra is a right ideal of R
containing aby the Proposition 3.2. This means that χRa is a fuzzy left ideal
and χaR∪Ra is a fuzzy right ideal of R, by the Lemma 2.4. By our supposition
χaR∪Ra ∩χRa ⊆χRa ◦χaR∪Ra,i.e., χ(aR∪Ra)∩Ra ⊆χ(Ra)(aR∪Ra).Thus (aR ∪
Ra)∩Ra ⊆Ra(aR ∪Ra).Since a∈(aR ∪Ra)∩Ra, i.e., a∈Ra(aR ∪Ra) =
(Ra)(aR)∪(Ra)(Ra).Now
(Ra)(aR)=(Ra)((ea)(RR)) = (Ra)((RR)(ae))
= (Ra)(((ae)R)R) = (Ra)((aR)R)
= (Ra)((RR)a) = (Ra)(Ra).
This implies that
(Ra)(aR)∪(Ra)(Ra)=(Ra)(Ra)∪(Ra)(Ra)
= (Ra)(Ra) = ((Ra)a)R
= ((Ra)(ea))R= ((Re)(aa))R
= (Ra2)R.
Thus a∈(Ra2)R, i.e., ais an intra regular. Therefore Ris an intra-regular,
i.e., (2) ⇒(1) .
Theorem 11. Let Rbe an intra-regular locally associative LA-ring. Then for
every fuzzy right ideal µof R, µ(an) = µ(a2n)for all a∈R, where nis any
positive integer.
Proof. For n= 1.Let a∈R, this implies that there exist elements xi, yi∈R
such that a=n
i=1(xia2)yi.Thus
µ(a) = µ((xia2)yi)≥µ(xia2)≥µ(a2)
=µ(aa)≥min{µ(a), µ (a)}=µ(a).
⇒µ(a) = µ(a2).
762 N. KAUSAR, M. MUNIR, B. UL ISLAM, M. ALESEMI, SALAHUDDIN and M. GULZAR
Now a2=aa = ((xia2)yi)((xia2)yi) = (x2
ia4)y2
i,then the result is true for
n= 2.Assume that the result is true for n=k, i.e., µ(ak) = µ(a2k).Now
ak+1 =aka= ((xk
ia2k)yk
i)((xia2)yi) = (xk+1
ia2(k+1))yk+1
i.Thus
µ(ak+1) = µ((xk+1
ia2(k+1))yk+1
i)≥µ(xk+1
ia2(k+1))
≥µ(a2(k+1)) = µ(ak+1ak+1)
≥min{µa(k+1), µ a(k+1)}=µa(k+1).
⇒µ(ak+1) = µ(a2(k+1) ).
Hence by induction method, the result is true for all positive integers.
Proposition 3.9. Let Rbe an intra-regular locally associative LA-ring with left
identity e. Then for every fuzzy right ideal µof R, µ(ab) = µ(ba)for all a, b ∈R.
Proof. Same as Lemma 3.9.
References
[1] R. J. Cho, J. Jezek and T. Kepka, Paramedial groupoids, Czechoslovak
Math. J., 49 (1999), 277-290.
[2] P. S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appli., 84
(1981), 264-269.
[3] K. C. Gupta and M. K. Kantroo, The intrinsic product of fuzzy subsets of
a ring, Fuzzy sets and systems, 57 (1993), 103-110.
[4] J. Jezek and T. Kepka, Medial groupoids, Rozpravy CSAV Rada mat. a
prir. ved 93/2, 1983, 93 pp.
[5] M. S. Kamran, Conditions for LA-semigroups to resemble associative struc-
tures, Ph.D. Thesis, Quaid-i-Azam University, Islamabad, 1993.
[6] M. A. Kazim, M. Naseerdin, On almost semigroups, Alig. Bull. Math., 2
(1972), 1-7.
[7] N. Kausar, A. Waqar, Characterizations of non-associative rings by their
intuitionistic fuzzy bi-ideals, European Journal of Pure and Applied Math-
ematics, 12 (2019), 226-250.
[8] N. Kausar, Direct product of finite intuitionistic fuzzy normal subrings over
non-associative rings, European Journal of Pure and Applied Mathematics,
12 (2019), 622-648.
[9] N. Kuroki, Fuzzy bi-ideals in semigroups, Comment. Math. Univ. St. Pauli,
28 (1979), 17-21.
FUZZY BI-IDEALS IN LA-RINGS 763
[10] N. Kuroki, Regular fuzzy duo rings, Inform. Sci., 94 (1996), 119-139.
[11] W. J. Liu, Fuzzy invariant subgroups and ideals, Fuzzy sets and systems, 8
(1982), 133-139.
[12] T. K. Mukherjee and M. K. Sen, On fuzzy ideals of a ring 1, Fuzzy sets
and systems, 21 (1987), 99-104.
[13] T. K. Mukherjee and M. K. Sen, Prime fuzzy ideals in rings, Fuzzy sets
and systems, 32 (1989), 337-341.
[14] P. V. Protic and N. Stevanovic, AG-test and some general properties of
Abel-Grassmann’s groupoids, Pure Math. Appli., 6 (1995), 371-383.
[15] V. S. Ramamurthi, Weakly regular rings, Canad. Math. Bull., 16 (1973),
317-321.
[16] T. Shah and I. Rehman, On LA-rings of finitely non-zero functions, Int. J.
Contempt. Math. Sci., 5 (2010), 209-222.
[17] T. Shah, N. Kausar, I. Rehman, Intuitionistic fuzzy normal subrings over a
non-associative ring, An. St. Univ. Ovidius Constanta, 20 (2012), 369-386.
[18] U. M. Swamy and K. L. N. Swamy, Fuzzy prime ideals of rings, J. Math.
Anal. Appl., 134 (1988), 94-103.
[19] L. A. Zadeh, Fuzzy sets, Information and control, 8 (1965), 338-363.
Accepted: 16.05.2019
