ORIGINAL PAPER

Equiprime, 3-prime and c-prime fuzzy ideals of nearrings

Babushri Srinivas Kedukodi Æ Syam Prasad Kuncham ÆSatyanarayana Bhavanari

Published online: 24 September 2008

� Springer-Verlag 2008

Abstract In this paper, we present the notions of equi-

prime fuzzy ideal, 3-prime fuzzy ideal and c-prime fuzzy

ideal of a nearring. We characterize these fuzzy ideals

using level subsets and fuzzy points. If f: N ? M is an onto

nearring homomorphism, we show that the map l 7! f ðlÞdefines a one-to-one correspondence between the set of all

f-invariant (alternatively with sup property) equiprime

(3-prime and c-prime, respectively) fuzzy ideals of N and

the set of all equiprime (3-prime and c-prime, respectively)

fuzzy ideals of M. Finally, we define fuzzy cosets deter-

mined by generalized fuzzy ideals; obtain fundamental

results and isomorphism theorems.

Keywords Nearring � Equiprime � 3-prime � c-Prime �Fuzzy ideal � Fuzzy point � Quasi-coincident � Level set �Fuzzy coset

1 Introduction

Mukherjee and Sen (1987) introduced the notion of fuzzy

ideal of a ring and Abou-Zaid (1991) studied fuzzy ideal of

a nearring. Although these definitions were fundamentally

important and had applications, they yielded examples

which lacked the fuzzy character. This was primarily due

to the characterization that a fuzzy subset l of a nearring N

is a fuzzy ideal of N if and only if the level subset lt =

{x [ N | l(x) C t} is an ideal of N for all t [ [0, 1]. This

problem was resolved by Bhakat and Das (1996) in rings

by using the idea of quasi-coincidence of a fuzzy point with

a fuzzy set. Bhakat and Das (1996) introduced a different

type of fuzzy subring (ideal) called (e, e _ q)-fuzzy subring

(ideal) which generalized the ordinary fuzzy ideal defined

by Mukherjee and Sen (1987). Davvaz (2006) extended the

(e, e _ q)-fuzzy subring (ideal) definition to nearrings.

Yuan et al. (2003) gave the definition of a fuzzy sub-

group with a lower threshold a and an upper threshold b.

Davvaz (2006) used this idea and further generalized the

concept of a ð�; � _ qÞ fuzzy ideal to a fuzzy ideal with

thresholds a and b. When a = 0 and b = 1 we get the

ordinary fuzzy ideal given by Abou-Zaid (1991) and when

a = 0 and b = 0.5 we get the (e, e _ q)-fuzzy ideal defined

by Davvaz (2006) (Bhakat and Das 1996 for rings).

Obviously, the primary benefit of the concept of thresholds

is the choice for thresholds which gives rise to the fuzzy

character in the examples. This motivates us to use the

concept of thresholds and study the notions of equiprime

fuzzy ideal, 3-prime fuzzy ideal and c-prime fuzzy ideal of

a nearring.

Booth et al. (1990) proved that every equiprime near-

ring N is zerosymmetric. That is, if {0} is an equiprime

ideal of N then a0 = 0 for every a [ N. In this paper, we

prove that for an ordinary (thresholds a = 0 and b = 1)

equiprime fuzzy ideal l of N, l(a0) = l(0) for every

a [ N. Also we show that this result need not hold for a

generalized equiprime fuzzy ideal l of N. However the

rationale in citing the above results is to note that even after

the process of fuzzy generalization, whenever required we

B. S. Kedukodi (&) � S. P. Kuncham

Department of Mathematics, Manipal Institute of Technology,

Manipal University, Manipal, Karnataka 576104, India

e-mail: babushrisrinivas@yahoo.co.in

S. P. Kuncham

e-mail: drkuncham@yahoo.com

S. Bhavanari

Department of Mathematics, Acharya Nagarjuna University,

Nagarjuna Nagar, Guntur, Andhra Pradesh 522510, India

e-mail: bhavanari2002@yahoo.co.in

123

Soft Comput (2009) 13:933–944

DOI 10.1007/s00500-008-0369-x

can simulate easily the vast literature of crisp algebra by

picking out appropriate thresholds.

2 Definitions and preliminaries

In this section, we review some definitions and results for the

sake of completeness. Throughout the paper, N and M will

denote right nearrings. A standard reference on nearrings is

Pilz (1983) and on rings is Anderson and Fuller (1992).

An ideal I of N is called equiprime if a 2 NnI and x, y

[ N with arx - ary [ I for all r [ N implies x - y [ I. N is

called an equiprime nearring if {0} is an equiprime ideal of

N. The equiprime radical of N is given by PeðNÞ ¼TfIjI is an equiprime ideal of Ng and the equiprime rad-

ical of an ideal I is given by <eðIÞ ¼TfPjP

is an equiprime ideal of N and I � Pg: (Booth et al. 1990)

An ideal I of N is called 3-prime if a, b [ N and anb [ I

for all n [ N implies a [ I or b [ I. N is called 3-prime

nearring if {0} is a 3-prime ideal of N. The 3-prime radical

of N is given by P3ðNÞ ¼TfIjI is a 3-prime ideal of Ng

and the 3-prime radical of an ideal I is given by <3ðIÞ ¼TfPjP is a 3-prime ideal of N and I � Pg: (Groenewald

1991)

An ideal I of N is called c-prime if a, b [ N and ab [ I

implies a [ I or b [ I . N is called c-prime nearring if {0} is

a c-prime ideal of N. The c-prime radical of N is given

by PcðNÞ ¼TfIjI is a c-prime ideal of Ng and the

c-prime radical of an idealI is given by <cðIÞ ¼TfPjP is a c-prime ideal of N and I � Pg: (Groenewald

1988)

The following implications are well known in nearrings

(Veldsman 1992; Groenewald 1988, 1991).

1. equiprime ideal ) 3-prime ideal.

2. c-prime ideal ) 3-prime ideal.

The notions of equiprime ideal, 3-prime ideal and prime

ideal coincide in rings. In commutative rings, all the above

notions coincide.

An ideal I of N is said to have insertion of factors

property (IFP) if a, b [ N and ab [ I implies anb [ I for all

n [ N. N is called integral if it has no nonzero zero divisors

(Pilz 1983).

For a nonempty set X, the mapping l: X ? [0, 1] is

called a fuzzy subset of X. Let a, b [ [0, 1] and a\ b.

Let l be a fuzzy subset of a near-ring N. Then l is called a

fuzzy ideal with thresholds of N, if for all x, y, i [ N,

(i) a _ lðxþ yÞ� b ^ lðxÞ ^ lðyÞ;(ii) a _ lð�xÞ� b ^ lðxÞ;(iii) a _ lðyþ x� yÞ� b ^ lðxÞ;(iv) a _ lðxyÞ� b ^ lðxÞ;(v) a _ lðxðyþ iÞ � xyÞ� b ^ lðiÞ (Davvaz 2006).

We can combine the axioms (i) and (ii) as follows:

a _ lðx� yÞ� b ^ lðxÞ ^ lðyÞ:

We call a as the lower threshold of N and b as the upper

threshold of N. Throughout this paper, a denotes the lower

threshold of N and b denotes the upper threshold of N.

The study of implication operators using the concept of

thresholds can be found in Davvaz (2008).

If l is a fuzzy subset of N, then for t [ [0, 1] the set lt

= {x [ N |l(x) C t} is called a level subset of N with

respect to l. We denote l* = {x [ N | l(x) C l(0)}.

Let f: N ? M be a mapping and l, r be fuzzy ideals of

N and M, respectively. Then the image of l under f is

defined as f ðlÞðyÞ ¼ supx2f�1ðyÞlðxÞ if f-1(y) = / and

f(l)(y) = 0 if f-1(y) = /. The preimage of r under f,

denoted by f-1(r), is the fuzzy subset of N defined by

f-1(r)(x) = r(f(x)) for all x [ N. l is called f-invariant if

f(x) = f(y) implies l(x) = l(y) for all x, y [ N.

A fuzzy ideal l is said to have the sup property if every

nonempty subset of Im(l) has a maximal element (Bhav-

anari et al. 2004).

An ordinary fuzzy ideal l of N is said to have IFP if for

every a, b [ N, l(anb) C l(ab) for all n [ N.

The study of ordinary prime fuzzy ideals can be found in

Kedukodi et al. (2007).

Let x [ N and t [ (0, 1] be fixed. A fuzzy subset l of N

of the form

lðyÞ ¼ t if y ¼ x0 if y 6¼ x

�

is said to be a fuzzy point with support x and value t and is

denoted by xt (Ming and Ming 1980).

Remark 2.1 Let l be a fuzzy ideal and b [ (0, 1].

Then lb 6¼ ; , lð0Þ� b

Proof As lb 6¼ ;; let x [ lb. Then we have

a _ lð0Þ ¼ a _ lðx� xÞ� b ^ lðxÞ ^ lðxÞ) a _ lð0Þ� b) lð0Þ� b:

The converse is straightforward.

In this sequel, fuzzy ideal means fuzzy ideal with

thresholds and l(0) C b.

3 Equiprime, 3-prime and c-prime fuzzy ideals

Definition 3.1 A fuzzy ideal l of N is called equiprime if

for all x, y, a [ N,

a _ lðaÞ _ lðx� yÞ� b ^ infr2N

lðarx� aryÞ:

Definition 3.2 A fuzzy ideal l of N is called 3-prime if

for all a, b [ N,

a _ lðaÞ _ lðbÞ� b ^ infn2N

lðanbÞ:

934 B. S. Kedukodi et al.

123

Definition 3.3 A fuzzy ideal l of N is called c-prime if

for all a, b [ N,

a _ lðaÞ _ lðbÞ� b ^ lðabÞ:

Example 3.4 Consider Z12 ¼ f�0; �1; �2; . . .; �11g (Ring of

integers modulo 12). Let a, b [ (0.2, 0.8) and a = b.

Define a fuzzy subset l: Z12? [0, 1] by

lðxÞ ¼

0:9 if x ¼ �60:8 if x ¼ �0a if x 2 f�3; �9gb if x 2 f�2; �4; �8; �10g0:2 elsewhere

8>>>><

>>>>:

Take thresholds a ¼ a ^ b and b ¼ a _ b: Then l is equi-

prime, 3-prime as well as c-prime fuzzy ideal of Z12. Note

that l0:9 ¼ f�6g is not an ideal of Z12.

Example 3.5 Let Z be the ring of integers. Define a fuzzy

subset l: Z ? [0, 1] by

lðxÞ ¼

0:99 if 12jx and x� 0

0:98 if 12jx and x\0

0:95 if 6jx and 12-xln 33

if 3jx and 6-xln jxjjxj if jxj is an odd prime1

ln 16if x 2 f�1; 1g

minf 1ln 16

; 1ln jxjg elsewhere

8>>>>>>>>><

>>>>>>>>>:

Then l is equiprime, 3-prime as well as c-prime fuzzy

ideal of Z for the thresholds a ¼ ln 55

and b ¼ ln 33: How-

ever, l is just a fuzzy ideal of Z for the thresholds a ¼ ln 55

and b = 0.8. Note that the cardinality of the image l is

infinity.

Since prime numbers play an important role in public

key cryptography, there is a great deal of interest in

determining whether a large number is prime or not. It is

well known that the expressionln jxjjxj may be regarded

(fuzzy!) as the probability that a positive integer picked at

random from among the first |x| integers will be prime. The

inverse of this probability gives approximately the number

of prime integers before |x|. Also, 1ln jxj gives approximately

the density of primes from the first |x| positive integers.

Example 3.6 Let N = {0, a, b, c} be a set with binary

operations ? and � defined as follows:

Then (N, ?, �) is a nearring. Define a fuzzy subset l by

lðxÞ ¼0:2 if x 2 fb; cg0:7 if x ¼ 0

0:9 if x ¼ a

8<

:

Take thresholds a = 0.3 and b = 0.6. Then l is 3-prime as

well as c-prime fuzzy ideal N. However, l is not an

equiprime fuzzy ideal N since

0:3 _ lðbÞ _ lðb� 0Þ ¼ 0:3l0:6 ¼ 0:6 ^ lð0Þ¼ 0:6 ^ inf

r2Nlðbrb� br0Þ:

Example 3.7 Let R be the ring of 2 9 2 matrices over

Z2 = {0, 1 }. Let

�x0 ¼0 0

0 0

� �

; �x1 ¼0 1

0 0

� �

; �x2 ¼1 0

0 0

� �

; �x3 ¼0 0

1 0

� �

;

�x4 ¼0 0

0 1

� �

and �x5 ¼1 1

1 1

� �

:

Let I ¼ f�x0g; J ¼ f�x1; �x2; �x3; �x4g and K ¼ f�x5g: Define a

fuzzy subset l by

lðxÞ ¼

0:1 if x 2 RnðI [ J [ KÞ0:2 if x 2 J0:6 if x 2 K0:9 if x 2 I

8>><

>>:

Then l is equiprime as well as 3-prime fuzzy ideal N for

the thresholds a = 0.7 and b = 0.8. But with these

thresholds l is not a c-prime fuzzy ideal of N since

0:7 _ lð�x1Þ _ lð�x2Þ ¼ 0:7l0:8 ¼ 0:8 ^ lð�x1�x2Þ:

Example 3.8 Let N = {0, a, b, c} be a set with binary

operations ? and � defined as follows:

? 0 a b c

0 0 a b c

a a 0 c b

b b c 0 a

c c b a 0

� 0 a b c

0 0 0 0 0

a 0 a 0 a

b b b b b

c b c b c

? 0 a b c

0 0 a b c

a a 0 c b

b b c 0 a

c c b a 0

Equiprime, 3-prime and c-prime fuzzy ideals of nearrings 935

123

Then (N, ? , �) is a nearring. Define a fuzzy subset l by

lðxÞ ¼0:3 if x 2 fb; cg0:8 if x ¼ 0

0:1 if x ¼ a

8<

:

Take thresholds a = 0.4 and b = 0.6. Then l is a

3-prime fuzzy ideal N. Note that l is not an equiprime

fuzzy ideal N since

0:4 _ lðaÞ _ lðc� aÞ ¼ 0:4l0:6 ¼ 0:6 ^ lð0Þ¼ 0:6 ^ inf

r2Nlðarc� araÞ:

Also l is not a c-prime fuzzy ideal of N since

0:4 _ lðcÞ _ lðaÞ ¼ 0:4l0:6 ¼ 0:9 ^ lðcaÞ:

Lemma 3.9 Let l be an equiprime (resp. 3-prime) fuzzy

ideal of N.

(i) For x, y, a [ N, if l(arx - ary) C b (resp. l(arx)

C b) V r [ N, then l(a) C b or l(x - y) C b (resp.

l (x) C b).

(ii) Let l be an ordinary equiprime fuzzy ideal. For x, y,

a [ N, if l(arx - ary) = l(0) for all r [ N, then -

l(a) = l(0) or l(x) = l(y).

(iii) For a, b [ N, l(a0) C b and a _ lða� b0Þ� lðaÞ ^b: If l is an ordinary equiprime fuzzy ideal,

then l(a0) = l(0) and l(a - b0) = l(a).

(iv) If there exist a, b [ N such that aN = b then lðbÞ� b:

Proof The proofs of (i) and (ii) are clear. To prove (iii),

consider

a _ lða0Þ _ lða0� 0Þ� b ^ inf

r2Nlðða0Þrða0Þ � ða0Þr0ÞÞ

¼ b ^ lð0Þ ¼ b:

This gives l(a0) C b. Now

a _ lða� b0Þ� b ^ lðaÞ ^ lðb0Þ¼ lðaÞ ^ ðb ^ lðb0ÞÞ ¼ lðaÞ ^ b:

This implies a _ lða� b0Þ� lðaÞ ^ b:Now let a = 0 and b = 1. In a contrary way we

assume there exists a [ N such that l(a0) = l(0). Note

that l(0) = l((a0)r(a0) - (a0)r0) for all r [ N. Using (ii),

we get l(a0) = l(0), a contradiction. The proof of

l(a - b0) = l(a) is straightforward.

To prove (iv), suppose there exist a, b [ N such that

aN = b. Now, (iii) gives b B l(a0) = l(b).

Now we give an example to show that l(a0) = l(0)

need not hold in general for an equiprime fuzzy ideal l.

Example 3.10 Let N be the nearring defined in Example

3.6.

Define a fuzzy subset l by

lðxÞ ¼0:2 if x 2 fa; cg0:6 if x ¼ b0:7 if x ¼ 0

8<

:

Take thresholds a = 0.4 and b = 0.6. Then l is an

equiprime fuzzy ideal N. However

lðb0Þ ¼ lðbÞ ¼ 0:6 6¼ 0:7 ¼ lð0Þ:

Theorem 3.11 Let l be a fuzzy ideal of N. Then l is an

equiprime (3-prime and c-prime, respectively) fuzzy ideal

of N if and only if for every t [ (a, b], the level subset lt

is an equiprime (3-prime and c-prime, respectively) ideal

of N.

Proof Note that by Theorem 3.11 of Davvaz (2008), l is

a fuzzy ideal of N if and only if for every t [ (a, b], the

level subset lt is an ideal of N. Let l be an equiprime

fuzzy ideal of N. Take t [ (a, b], x, y, a [ N such that

arx - ary [ lt for all r [ N. This implies l(arx - ary) C t

for all r [ N. Hence

infr2N

lðarx� aryÞ� t:

As l is an equiprime fuzzy ideal of N, we get

a _ lðaÞ _ lðx� yÞ� b ^ infr2N

lðarx� aryÞ� b ^ t ¼ t:

Hence we get a [ lt or x - y [ lt. Thus lt is an equiprime

fuzzy ideal of N.

To prove the converse, if possible assume that there

exist a, x, y [ N such that

a _ lðaÞ _ lðx� yÞ\b ^ infr2N

lðarx� aryÞ:

Choose t such that

a _ lðaÞ _ lðx� yÞ\t\b ^ infr2N

lðarx� aryÞ:

This implies l(a) \ t, l(x - y) \ t and

infr2N

lðarx� aryÞ[ t:

This further implies a 62 lt; x� y 62 lt and arx - ary [ lt

for all r [ N. This is a contradiction to the assumption

that lt is an equiprime fuzzy ideal of N for every t [ (a, b].

We can similarly prove the other results.

� 0 a b c

0 0 0 0 0

a a a a a

b 0 a b c

c a 0 c b

936 B. S. Kedukodi et al.

123

Corollary 3.12 Let N be a nearring with a set of equi-

prime (3-prime and c-prime, respectively) ideals I0 , I1

,_, In = N of N. Then there exists an equiprime (resp.

3-prime, c-prime) fuzzy ideal l of N whose level ideals are

precisely the members of the chain with lb = I0.

Proof Let {ti; ti [ (a, b); i = 1, 2,..., n} be such that

t1 [ t2 [_[ tn. Choose t [ [0, 1] with t C b. Define

l: N ? [0, 1] as follows:

lðxÞ ¼

t if x 2 I0;t1 if x 2 I1nI0;: :: :: :tn if x 2 InnIn�1:

8>>>>>><

>>>>>>:

Then l is an equiprime (3-prime and c-prime, respectively)

fuzzy ideal of N whose level ideals are precisely the

members of the chain and lb = I0.

Theorem 3.13 Every equiprime fuzzy ideal of N is a

3-prime fuzzy ideal of N. The converse holds if N is a ring.

Proof Let l be an equiprime fuzzy ideal of N and

a, b [ N. By Lemma 3.9 (iii), we get

infn2N

lðanb� an0Þ� infn2N

lðanbÞ:

Since l is an equiprime fuzzy ideal of N,

a_lðaÞ _ lðb� 0Þ� b ^ infn2N

lðanb� an0Þ:

) a _ lðaÞ _ lðbÞ� b ^ infn2N

lðanbÞ:

Thus l is a 3-prime fuzzy ideal of N. To prove the

converse, let N be a ring and x, y, a [ N. As l is a 3-prime

fuzzy ideal of N, we have

a _ lðaÞ _ lðx� yÞ� b ^ infr2N

lðarðx� yÞÞ

¼ b ^ infr2N

lðarx� aryÞ:

Thus l is an equiprime fuzzy ideal of N.

Remark 3.14 Alternatively, we can prove Theorem 3.13

by using the Theorem 3.11 and the fact that every equi-

prime ideal of N is a 3-prime ideal of N and both these

concepts coincide in a ring (refer Veldsman 1992).

Theorem 3.15 Every c-prime fuzzy ideal of N is a

3-prime fuzzy ideal of N. The converse holds if N is a

commutative ring.

Theorem 3.16 Let R be a commutative ring. Then the

following are equivalent:

(i) l is an equiprime fuzzy ideal of R.

(ii) l is a 3-prime fuzzy ideal of R.

(iii) l is a c-prime fuzzy ideal of R.

Remark 3.17 The fuzzy ideal l as defined in Example 3.8

shows that the converse of the Theorem 3.13 and that of

Theorem 3.13 are not true in general. Also Example 3.7

shows that Theorem 3.16 does not hold if R is not a

commutative ring.

Definition 3.18 A fuzzy ideal l of N is said to have

insertion of factors property (IFP) if for every a; b 2N; a _ lðanbÞ� b ^ lðabÞ for all n [ N.

Example 3.19 The fuzzy ideal l defined in Example 3.4

has the IFP.

Theorem 3.20 Let l be a fuzzy ideal of N. Then the

following are equivalent:

(i) l has the IFP.

(ii) lt has the IFP for all t [ (a, b].

Proof [(i) ) (ii)] Let t [ (a, b] and a, b [ N such that

ab [ lt. Take n [ N. As l has the IFP,

a _ lðanbÞ� b ^ lðabÞ� b ^ t ¼ t ðas t 2 ða; b�Þ) lðanbÞ� t ðas t 2 ða; b�Þ) anb 2 lt:

Hence lt has the IFP for all t [ (a, b].

[(ii) ) (i)] If possible suppose that l does not have the

IFP. Then there exist a, n, b [ N such that

a _ lðanbÞ\b ^ lðabÞ:

Choose t [ (a, b) such that

a _ lðanbÞ\t\b ^ lðabÞ) lðanbÞ\t\lðabÞ) anb 62 lt and ab 2 lt:

This is a contradiction to the fact that lt has the IFP for all

t [ (a, b]. Thus l has the IFP.

Theorem 3.21 Let l be an equiprime fuzzy ideal of N

with IFP. Then l is a c-prime fuzzy ideal of N.

Proof Let t [ (a, b] and a, b [ N such that ab [ lt. As lhas the IFP, by Theorem 3.20 we have

anb 2 lt 8 n 2 N:

Now by Theorem 3.11, lt is an equiprime ideal of N. As

the constant part of N is contained in any equiprime ideal

of N, anb - an0 [ lt for all n [ N.

) a 2 lt or ðb� 0Þ ¼ b 2 lt:

This proves that l is a c-prime fuzzy ideal of N.

Remark 3.22 Consider the fuzzy ideal l defined in

Example 3.7. We have

0:7 _ lð�x1 �x5 �x2Þ ¼ 0:7\0:8 ¼ 0:8 ^ lð�x1 �x2Þ:

Equiprime, 3-prime and c-prime fuzzy ideals of nearrings 937

123

This implies l does not have the IFP. Note that l is an

equiprime fuzzy ideal of N but not a c-prime fuzzy ideal

of N.

Proposition 3.23. Let f: N ? M be a (resp. onto, onto)

homomorphism. If l is a (resp. equiprime, 3-prime) fuzzy

ideal of M, then f-1(l) is a (resp. equiprime, 3-prime) fuzzy

ideal of N with the same thresholds as that of l.

Proof Let x, y, a [ N. Consider

a _ f�1ðlÞðx� yÞ ¼ a _ lðf ðx� yÞÞ¼ a _ lðf ðxÞ � f ðyÞÞ� b ^ lðf ðxÞÞ ^ lðf ðyÞÞ¼ b ^ f�1ðlÞðxÞ ^ f�1ðlÞðyÞ:

a _ f�1ðlÞðyþ x� yÞ ¼ a _ lðf ðyþ x� yÞÞ¼ a _ lðf ðyÞ þ f ðxÞ � f ðyÞÞ� b ^ lðf ðxÞÞ ¼ b ^ f�1ðlÞðxÞ:

a _ f�1ðlÞðxyÞ ¼ a _ lðf ðxyÞÞ ¼ a _ lðf ðxÞf ðyÞÞ� b ^ lðf ðxÞÞ ¼ b ^ f�1ðlÞðxÞ:

a _ f�1ðlÞððxþ aÞy� xyÞ¼ a _ lððf ðxÞ þ f ðaÞÞf ðyÞ � f ðxÞf ðyÞÞ� b ^ lðf ðaÞÞ¼ b ^ f�1ðlÞðaÞ:

This proves that f-1(l) is a fuzzy ideal of N.

Now let t [ (a, b], a, x, y [ N such that arx - ary

[ (f-1(l))t for all r [ N. Then f-1(l)(arx - ary) C t for all

r [ N. This implies l(f(a) f(r) f (x) - f (a) f(r) f (y)) C t for

all f(r) [ M. Hence f(a) f(r) f (x) - f (a) f(r) f (y) C lt for

all f(r) [ M. By Theorem 3.11, lt is an equiprime fuzzy

ideal of M. Note that since f is onto, any r0 [ M is of the

form r0 = f(r) for some r [ N. Hence we get f(a) [ lt or

f(x) - f(y) [ lt. Thus a [ f-1(lt) = (f-1(l))t or x - y [f-1(lt) = (f-1(l))t. Again by Theorem 3.11, f-1(l) is an

equiprime fuzzy ideal of N.

The other result can be proved similarly.

Proposition 3.24 Let f: N ? M be a homomorphism. If lis a c-prime fuzzy ideal of M, then f-1(l) is a c-prime fuzzy

ideal of N with the same thresholds as that of l.

Proposition 3.25 Let f: N ? M be a homomorphism. If lis a fuzzy ideal of N, then f(l) is a fuzzy ideal of f(N) with

the same thresholds as that of l.

Now we give an example to illustrate that usually the

notions of equiprime, 3-prime and c-prime fuzzy ideals are

not preserved under homomorphism images.

Example 3.26 Let Z be set of integers and Z8 ¼f�0; �1; �2; . . .; �7g (Ring of integers modulo 8). Let f: Z ? Z8

be any onto homomorphism (example f(x) = x mod8).

Define l: Z ? [0, 1] by

lðxÞ ¼ 1 if n ¼ 0

0 if n 6¼ 0

�

Then l is an equiprime fuzzy ideal of Z with thresholds

a = 0 and b = 1. We have

f ðlÞð�nÞ ¼ 1 if �n ¼ �00 if �n 6¼ �0

�

Hence f ðlÞð�4 �n �2� �4 �n �4Þ ¼ 1 for all �n 2 Z8 and f ðlÞð�4Þ ¼0 ¼ f ðlÞð�2� �4Þ: As �4 6¼ �0 and �2� �4 6¼ �0; f(l) is not an

equiprime fuzzy ideal of Z8 with thresholds a = 0 and

b = 1.

We now find the conditions under which the equiprime,

3-prime and c-prime fuzzy ideals are preserved under

homomorphism images.

Proposition 3.27 Let f: N ? M be an onto nearring

homomorphism and let l be a f-invariant fuzzy ideal of N .

For t [ (a, b], if lt is an equiprime (resp. 3-prime, c-prime)

ideal of N then f(lt) is an equiprime (resp. 3-prime,

c-prime) ideal of M.

Theorem 3.28 Let f: N ? M be an onto nearring

homomorphism and l be a fuzzy subset of N. If l is a

f-invariant equiprime (resp. 3-prime, c-prime) fuzzy ideal

of N then f(l) is an equiprime (resp. 3-prime, c-prime)

fuzzy ideal of M with the same thresholds as that of l.

Proof Using Proposition 3.27, f(lt) is an equiprime fuzzy

ideal of M for all t [ (a, b]. Since l is f-invariant and f is

onto, f(lt) = (f(l))t for all t [ (a, b]. An appeal to Theorem

3.11 concludes that f(l) is an equiprime fuzzy ideal of M.

We can similarly prove the other results.

Theorem 3.29 Let f: N ? M be an onto nearring

homomorphism and F be the set of all f-invariant equi-

prime (resp. 3-prime, c-prime) fuzzy ideals of N with the

lower threshold a and the upper threshold b. Then the map

l 7! f ðlÞ defines a one-to-one correspondence between F

and the set of all equiprime (resp. 3-prime, c-prime) fuzzy

ideals of M with the lower threshold a and the upper

threshold b.

Proof Clear from Proposition 3.24 and Theorem 3.28

Remark 3.30 Let f: N ? M be a homomorphism and lbe a fuzzy ideal of N. Then for t [ [0, 1], f ðltÞ � ðf ðlÞÞt:If l has the sup property then f(lt) = (f(l))t.

Theorem 3.31 Let f N ?M be a nearring homomorphism

and l be a fuzzy subset of N. If l is an equiprime (resp. 3-

prime, c-prime) fuzzy ideal of N with sup property then f(l)

is an equiprime (resp. 3-prime, c-prime) fuzzy ideal of M

with the same thresholds as that of l

Proof Using the Remark 3.30, the proof is similar to that

of Theorem 3.28.

938 B. S. Kedukodi et al.

123

Theorem 3.32 Let f: N ? M is an onto nearring homo-

morphism. Let S be the set of all equiprime (resp.

3-prime,c-prime) fuzzy ideals of N with sup property and

having the lower threshold a and upper threshold b. Then

the map l 7! f ðlÞ defines a one-to-one correspondence

between S and the set of all equiprime (resp. 3-prime,

c-prime) fuzzy ideals of M with the lower threshold a and

the upper threshold b.

Proof Clear from Proposition 3.24 and Theorem 3.31.

Definition 3.33 A fuzzy point xt is said to be quasi-

coincident with a fuzzy ideal l of N if a _l(x) ? t [ 2b.

This is denoted by xtq l. We say that xt belongs to ldenoted by xt [ l if a _ lðxÞ� t:

We denote ltq ¼ fx 2 N j xtqlg and

lt_q ¼ fx 2 N j xt 2 l or xtqlg¼ fx 2 N j a _ lðxÞ� t or a _ lðxÞ þ t [ 2bg:

By the notation xt [ lq we mean x 2 lt_q:

Theorem 3.34 Let l be a fuzzy subset of N. Then the

following are equivalent:

(a) l is a fuzzy ideal of N.

(b) For every t [ (0, 1], lt_q is an ideal of N.

Proof (a) ) (b) Let t [ (0, 1].

(i) Let x; y 2 lt_q: We claim x� y 2 lt_q:

We have a _ lðxÞ� t or a _ lðxÞ þ t [ 2b and a _lðyÞ� t or a _ lðyÞ þ t [ 2b:

As l is a fuzzy ideal of N, we have

a _ lðx� yÞ ¼ a _ a _ lðx� yÞ� a _ ðb ^ lðxÞ ^ lðyÞÞ¼ ða _ bÞ ^ ða _ lðxÞÞ ^ ða _ lðyÞÞ¼ b ^ ða _ lðxÞÞ ^ ða _ lðyÞÞ:

Case 1: Suppose a _ lðxÞ� t and a _ lðyÞ� t: Then a _lðx� yÞ� b ^ t ^ t ¼ b ^ t: If b ^ t ¼ t then x� y 2 lt_q:

This proof holds if t = b. Hence assume t = b. If b ^ t ¼b then a _ lðx� yÞ� b and t [b. Then we get a _ lðx�yÞ þ t� bþ t [ bþ b ¼ 2b: Hence x� y 2 lt_q:

Case 2: Suppose a _ lðxÞ� t and a _ lðyÞ þ t [ 2b:Then a _ lðx� yÞ� b ^ t ^ ð2b� tÞ: If b ^ t ^ ð2b� tÞ ¼t then x� y 2 lt_q: This proof holds if t = b. Hence assume

t = b. If b ^ t ^ ð2b� tÞ ¼ b then a _ lðx� yÞ� b and

t [ b. Then we get a _ lðx� yÞ þ t� bþ t [ bþ b ¼ 2b:Hence x� y 2 lt_q: If b ^ t ^ ð2b� tÞ ¼ ð2b� tÞ then

t [ (2b - t) and a _ lðx� yÞ[ ð2b� tÞ: Now a _ lðx�yÞ þ t [ ð2b� tÞ þ t ¼ 2b: Hence x� y 2 lt_q:

Case 3: Suppose a _ lðxÞ þ t [ 2b and a _ lðyÞ� t:

We omit the proof as it is similar to case 2.

Case 4: Suppose a _ lðxÞ þ t [ 2b and a _ lðyÞ þt [ 2b: Then a _ lðx� yÞ� b ^ ð2b� tÞ ^ ð2b� tÞ ¼ b ^ð2b� tÞ: If b ^ ð2b� tÞ ¼ b then a _ lðx� yÞ� b and

(2b - t) C b. That is a _ lðx� yÞ� b and b C t. This

implies a _ lðx� yÞ� t: Hence x� y 2 lt_q: This proof

holds if t = b. Hence assume t = b. If b ^ ð2b� tÞ ¼ð2b� tÞ then a _ lðx� yÞ[ ð2b� tÞ: Now a _ lðx�yÞ þ t [ ð2b� tÞ þ t ¼ 2b: Hence x� y 2 lt_q: We can

similarly prove the following.

(ii) If x 2 lt_q; y [ N then yþ x� y 2 lt_q:

(iii) If x 2 lt_q; y [ N then xy 2 lt_q:

(iv) If i 2 lt_q; x,y [ N then xðyþ iÞ � xy 2 lt_q:

Using (i)–(iv), the level subset lt_q is an ideal of N.

(b) ) (a): We will prove (i) a _ lðx� yÞ� b ^ lðxÞ ^lðyÞ for all x, y [ N. If possible suppose that there exist

x, y [ N such that a _ lðx� yÞ\b ^ lðxÞ ^ lðyÞ: Choose

t [ (a, b) such that a _ lðx� yÞ\t\b ^ lðxÞ ^ lðyÞ: Note

that a _ lðx� yÞ\t and a _ lðx� yÞ þ t\t þ t\2b:This implies x� y 62 lt_q: As t\b ^ lðxÞ ^ lðyÞ; we

have l(x) [ t and l(y) [ t. This implies a _ lðxÞ[ t and

a _ lðyÞ[ t: Thus we get x; y 2 lt_q but x� y 62 lt_q: This

is a contradiction to the fact that lt_q is an ideal of N.

Similarly for x, y, i [ N, we can prove the following:

(ii) a _ lðyþ x� yÞ� b ^ lðxÞ(iii) a _ lðxyÞ� b ^ lðxÞ;(iv) a _ lðxðyþ iÞ � xyÞ� b ^ lðiÞ:

Using (i)–(iv), l is a fuzzy ideal of N.

Theorem 3.35 Let l be a fuzzy ideal of N. Then the

following are equivalent:

(a) l is an equiprime fuzzy ideal of N.

(b) For every t [ (0, 1], lt_q is an equiprime ideal of N.

Proof (a)) (b) Let t [ (0, 1]. Using Theorem 3.34, lt_q

is an ideal of N. Let arx� ary 2 lt_q for all r in N. We

claim a 2 lt_q or x� y 2 lt_q: As l is an equiprime fuzzy

ideal of N, we have

a _ lðaÞ _ lðx� yÞ� b ^ infr2N

lðarx� aryÞ:

) a _ a _ lðaÞ _ lðx� yÞ� b ^ ða _ infr2N

lðarx� aryÞÞ

) a _ lðaÞ _ lðx� yÞ� b ^ t ^ ð2b� tÞ:

Case 1: Suppose b ^ t ^ ð2b� tÞ ¼ t: Then we have a _lðaÞ� t or a _ lðx� yÞ� t: Hence a 2 lt_q or

x� y 2 lt_q:

The proof given in Case 1 holds for t = b. Hence

assume t = b in the following cases.

Case 2: Suppose b ^ t ^ ð2b� tÞ ¼ b: Then t [ b and

a _ lðaÞ _ lðx� yÞ� b: This implies a _ lðaÞ� b or a _lðx� yÞ� b: If a _ lðaÞ� b then a _ lðaÞ þ t� bþ

Equiprime, 3-prime and c-prime fuzzy ideals of nearrings 939

123

t [ bþ b ¼ 2b: Hence a 2 lt_q: If a _ lðx� yÞ� bthen a _ lðx� yÞ þ t� bþ t [ bþ b ¼ 2b: Hence x� y

2 lt_q:

Case 3: Suppose b ^ t ^ ð2b� tÞ ¼ ð2b� tÞ: Then

(2b - t) \ b and a _ lðaÞ _ lðx� yÞ� ð2b� tÞ (Due to

the involvement of infimum in the definition of an equiprime

fuzzy ideal, we have to consider the case a _ lðaÞ _ lðx�yÞ ¼ ð2b� tÞÞ: We have the following subcases.

(i) Consider a _ lðaÞ _ lðx� yÞ[ ð2b� tÞ: Then a _lðaÞ[ ð2b� tÞ or a _ lðx� yÞ[ ð2b� tÞ: This

implies a _ lðaÞ þ t [ ð2b� tÞ þ t ¼ 2b or a _lðx� yÞ þ t [ ð2b� tÞ þ t ¼ 2b: Hence a 2 lt_q or

x� y 2 lt_q:

(ii) Now consider a _ lðaÞ _ lðx� yÞ ¼ ð2b� tÞ: This

implies a _ lðaÞ ¼ ð2b� tÞ or a _ lðx� yÞ ¼ ð2b�tÞ: If b\ t then our assumption b ^ t ^ ð2b� tÞ ¼ð2b� tÞ gives b ^ ð2b� tÞ ¼ ð2b� tÞ: This implies

b C (2b - t). This gives the contradiction t C b.

Hence b[ t. Then we have a _ lðaÞ ¼ ð2b� tÞ[ t

or a _ lðx� yÞ ¼ ð2b� tÞ[ t: Hence a 2 lt_q or

x� y 2 lt_q:

Thus lt_q is an equiprime ideal of N.

(b)) (a): Using Theorem 3.34, l is a fuzzy ideal of N.

If possible assume that there exist a,x, y [ N such that

a _ lðaÞ _ lðx� yÞ\b ^ infr2Nlðarx� aryÞ:Choose t [ (a, b) such that

a _ lðaÞ _ lðx� yÞ\t\b ^ infr2N

lðarx� aryÞ:

) a _ lðaÞ\t; a _ lðx� yÞ\t;

a _ lðaÞ þ t\t þ t ¼ 2t\2b and

a _ lðx� yÞ þ t\t þ t ¼ 2t\2b:

Hence we get a 62 lt_q; x� y 62 lt_q:

Also; t\b ^ infr2N

lðarx� aryÞ

) lðarx� aryÞ[ t for all r 2 N

) a _ lðarx� aryÞ[ t for all r 2 N

) arx� ary 2 lt_q for all r 2 N:

This is a contradiction to the assumption that lt_q is an

equiprime fuzzy ideal of N for every t [ (0, 1]. This

completes the proof.

Theorem 3.36 Let l be a fuzzy ideal of N. Then the

following are equivalent:

(a) l is a 3-prime (resp. c-prime) fuzzy ideal of N.

(b) For every t [ (0, 1], lt_q is a 3-prime (resp. c-prime)

ideal of N.

Theorem 3.37 Let l be a fuzzy subset of N. Then l is a

fuzzy ideal of N if and only if the fuzzy points satisfy the

following conditions:

(a) xt; yr 2 l) ðx� yÞt^r 2 lq:

(b) xt [ l,y [ N ) (y ? x - y)t [ lq.

(c) xt [ l,y [ N ) (xy)t [ lq.

(d) it [ l,x, y [ N ) (x(y ? i) - xy)t [ lq. Further we

have the following:

(e) l is an equiprime fuzzy ideal of N if and only if

conditions (a)–(d) are satisfied and for each a, x, y

[ N,

ðarx� aryÞt 2 l 8 r 2 N ) at 2 lq or ðx� yÞt 2 lq:

(f) l is a 3-prime fuzzy ideal of N if and only if conditions

(a)–(d) are satisfied and for each x, y [ N,

ðxryÞt 2 l 8 r 2 N ) xt 2 lq or yt 2 lq:

(g) l is a c-prime fuzzy ideal of N if and only if conditions

(a)–(d) are satisfied and for each x, y [N,

ðxyÞt 2 l) xt 2 lq or yt 2 lq:

Proof (Necessary part) Let l be a fuzzy ideal of N and

suppose fuzzy points xt, yr such that xt, yr [ l. Then we

have a _ lðxÞ� t and a _ lðyÞ� r: As l is a fuzzy ideal of

N, we have a _ lðx� yÞ� b ^ ða _ lðxÞÞ ^ ða _ lðyÞÞ:Then we have a _ lðx� yÞ� b ^ t ^ r: If b ^ t ^ r ¼ t ^r then a _ lðx� yÞ� t ^ r: This implies ðx� yÞt^r 2 lq:

This proof holds for t ^ r ¼ b: Now assume t ^ r 6¼ b: If

b ^ t ^ r ¼ b then t ^ r [ b and a _ lðx� yÞ� b: Now

consider a _ lðx� yÞ þ ðt ^ rÞ[ bþ b ¼ 2b: Hence ðx�yÞt^r 2 lq: This proves (a). The proofs of (b) and (c) are

straightforward.

To prove (d), let it [ l, x, y [ N. Then we have a _lðiÞ� t: As l is a fuzzy ideal of N; we get a _ lðxðyþiÞ � xyÞ� b ^ ða _ lðiÞÞ� b ^ t: If b ^ t ¼ t then clearly

(x(y ? i) - xy)t [ lq. This proof holds for t = b. Hence

assume t = b. If b ^ t ¼ b then t [b and a _ lðxðyþiÞ � xyÞ� b: Consider a _ lðxðyþ iÞ � xyÞ þ t� bþt [ bþ b ¼ 2b: Hence we get (x(y ? i)- xy)t [ lq. This

proves (iv).

To prove (e), suppose a, x, y [ N such that (arx -

ary)t [ l V r [ N. This implies arx� ary 2 lt_q8 r 2 N:

By Theorem 3.35, lt_q is an equiprime ideal of N. Hence

a 2 lt_q or x - y [ lt _q. This implies at [ lq or (x - y)t

[ lq. We can similarly prove (f) and (g).

(Sufficient part) We will prove

(i) a _ l(x - y) C b ^ l(x)^ l(y) for all x, y [ N.

If possible suppose that there exist x, y [ N such that

a _ l(x - y) \ b ^ l(x) ^ l(y). Choose t [ (a, b) such

that a _ l(x - y) \ t \b ^ l(x) ^ l(y). Note that

a _ l(x - y) \ t and a _ l(x - y) ? t \ t ? t \ 2b.

This implies ðx� yÞt 62 lq: As t \ b ^ l(x) ^ l(y), we

have l(x) [ t and l(y) [ t. This implies a _ l(x) [ t

and a _ l(y) [ t. Thus we get xt, yt [ l but ðx� yÞt 62

940 B. S. Kedukodi et al.

123

lq: This is a contradiction to the assumption (a) (take

r = t).

Similarly for x, y, i [ N, we can prove the following:

(ii) a _ l(y ? x - y) C b ^ l(x)

(iii) a _ l(xy) C b ^ l(x),

(iv) a _ l(x (y ? i) - xy) C b ^ l(i).

Using (i)–(iv), l is a fuzzy ideal of N.

To prove the sufficient part of (e), if possible assume

that there exist a, x, y [ N such that

a _ lðaÞ _ lðx� yÞ\b ^ infr2Nlðarx� aryÞ:Choose t [ (a, b) such that

a _ lðaÞ _ lðx� yÞ\t\b ^ infr2N

lðarx� aryÞ:

) a _ lðaÞ\t; a _ lðx� yÞ\t;

a _ lðaÞ þ t\t þ t ¼ 2t\2b and

a _ lðx� yÞ þ t\t þ t ¼ 2t\2b:

Hence we get at 62 lq; ðx� yÞt 62 lq:

Also,

t\b ^ infr2N

lðarx� aryÞ

) lðarx� aryÞ[ t for all r 2 N

) a _ lðarx� aryÞ[ t for all r 2 N

) ðarx� aryÞt 2 lq for all r 2 N:

This is a contradiction to the assumption in (e). We can

similarly prove the sufficient parts of (f) and (g). This

completes the proof.

4 Fuzzy cosets and radicals

Definition 4.1 Let l be a fuzzy ideal of N and x [ N. The

fuzzy subset xl of N defined by xl(n) = (a _ l(n - x)) ^ bVn [ N is called the fuzzy coset determined by x and l.

Theorem 4.2 For any fuzzy ideal l of N, N/l, the set of

all fuzzy cosets of l in N is a nearring under the addition

and multiplication defined as follows: xlþ yl ¼xþyl; xl � yl ¼ x�yl 8x; y 2 N: Further, �l : N=l! ½0; 1�defined by �lðxlÞ ¼ lðxÞ 8xl 2 N=l is a fuzzy ideal

of N/l.

Proof We first show that the operations are well defined.

Let a, b, c, d [ N be such that al = bl and cl = dl. Then

ða _ lðn� aÞÞ ^ b ¼ ða _ lðn� bÞÞ ^ b 8 n 2 N; ð1Þða _ lðn� cÞÞ ^ b ¼ ða _ lðn� dÞÞ ^ b 8 n 2 N: ð2Þ

Put n = a in (1) and n = c in (2). Then we get

ða _ lða� bÞÞ ^ b ¼ ða _ lð0ÞÞ ^ b ¼ b; ð3Þða _ lðc� dÞÞ ^ b ¼ ða _ lð0ÞÞ ^ b ¼ b: ð4Þ

If n = a ? c - d, then from (1) we get

ða _ lðaþ c� d � bÞÞ ^ b

¼ ða _ lðaþ c� d � aÞÞ ^ b

¼ ða _ a _ lðaþ c� d � aÞÞ ^ b

� ½a _ ðb ^ lðc� dÞÞ� ^ b

¼ ½ða _ bÞ ^ ða _ lðc� dÞÞ� ^ b

� b ^ b ¼ b ðby ð4ÞÞ:) a _ lðaþ c� d � bÞ� b:

ð5Þ

Now let n [ N. We have

ðalþ clÞðnÞ ¼ aþclðnÞ ¼ ½a _ lðn� ðaþ cÞÞ� ^ b

¼ ½a _ lððn� d � bÞ � ðaþ c� d � bÞÞ� ^ b

¼ ½a _ a _ lððn� d � bÞ � ðaþ c� d � bÞÞ� ^ b

� ½a _ ðb ^ lðn� d � bÞ ^ lðaþ c� d � bÞÞ� ^ b

� ½ða _ bÞ ^ ða _ lðn� d � bÞÞ^ða _ lðaþ c� d � bÞÞ� ^ b

�ða _ lðn� d � bÞÞ ^ b ðby ð5ÞÞ¼ ½a _ lðn� ðbþ dÞÞ� ^ b

¼ bþdlðnÞ ¼ ðblþ dlÞðnÞ:

Hence al ?cl C bl ?dl. Similarly we can prove

bl ?dl C al ?cl. Thus al ?cl = bl ?dl. This proves

that the addition is well defined.

Now consider

ða _ lðac� bdÞÞ ^ b

¼ ða _ lðac� bcþ bc� bdÞÞ ^ b

¼ ½a _ a _ lðða� bÞcþ bðd þ ð�d þ cÞÞ � bdÞ� ^ b

� ½ða _ ðb ^ lðða� bÞcÞ^lðbðd þ ð�d þ cÞÞ � bdÞÞ� ^ b

¼ ½ða _ bÞ ^ ða _ lðða� bÞcÞÞ^ða _ lðbðd þ ð�d þ cÞÞ � bdÞÞ� ^ b

¼ ½b ^ ða _ a _ lðða� bÞcÞÞ^ða _ a _ lðbðd þ ð�d þ cÞÞ � bdÞÞ� ^ b

� ½b ^ ½ða _ ðb ^ lða� bÞÞ� ^ða _ ðb ^ lð�d þ cÞÞ� ^ b

¼ b ^ ½ða _ lða� bÞÞ ^ b� ^ ½ða _ lð�d þ cÞÞ ^ b�� b ^ b ^ ½ða _ lð�d þ cÞÞ ^ b� ðby ð3ÞÞ¼ b ^ ½ða _ a _ lð�d þ cÞÞ ^ b�¼ b ^ ½ða _ a _ lð�cþ ðc� dÞ þ cÞÞ ^ b�� b ^ ½ða _ ðb ^ lðc� dÞÞ ^ b�¼ b ^ ½ða _ lðc� dÞÞ ^ b�� b ^ b ¼ b ðby ð4ÞÞ:

Equiprime, 3-prime and c-prime fuzzy ideals of nearrings 941

123

Let n [ N. Consider

ðal � clÞðnÞ ¼ a�clðnÞ ¼ ½a _ lðn� ðacÞÞ� ^ b

¼ ½a _ lððn� bdÞ � ðac� bdÞÞ� ^ b

¼ ½a _ a _ lððn� bdÞ � ðac� bdÞÞ� ^ b

¼ ½a _ ðb ^ lðn� bdÞ ^ lðac� bdÞÞ� ^ b

¼ ½ða _ bÞ ^ ða _ lðn� bdÞÞ ^ ða _ lðac� bdÞÞ� ^ b

�ða _ lðn� bdÞÞ ^ b ¼ b�dlðnÞ ¼ ðbl � dlÞðnÞ:

Hence al �cl C bl �dl. Similarly we can prove bl �dlC al�cl. Thus al �cl = bl �dl. This proves that the multi-

plication is well defined.

It is now easy to verify that N/l is a nearring with 0l as

the zero element and -xl is the negative of xl for all x [ N.

Let x, y, i [ N. Now consider

a _ �lðxl� ylÞ ¼ a _ �lðx�ylÞ¼ a _ lðx� yÞ� b ^ lðxÞ ^ lðyÞ¼ b ^ �lðxlÞ ^ �lðylÞ:

a _ �lðxlþ yl� xlÞ ¼ a _ �lðxþy�xlÞ¼ a _ lðxþ y� xÞ� b ^ lðyÞ¼ b ^ �lðylÞ:

a _ �lðxlylÞ ¼ a _ �lðxylÞ¼ a _ lðxyÞ� b ^ lðxÞ¼ b ^ �lðxlÞ:

a _ �lðxlðylþ ilÞÞ � xlylÞ ¼ a _ �lðxðyþiÞ�xylÞ¼ a _ lðxðyþ iÞ � xyÞ� b ^ lðiÞ¼ b ^ �lðilÞ:

Thus �l is a fuzzy ideal of N/l.

Theorem 4.3 Let l be a fuzzy ideal of N. For every x, y

[ N, xl = yl , l(x - y) C b.

Proof Take x, y [ N. Let xl = yl. Then (a _ l(n - x)) ^b = (a _ l(n - y)) ^ b V n [N. Put n = x. Then (a_ l(x - y))^ b = (a _ l(0))^ b = b ) a _ l(x - y) C b) l(x - y) C b. To prove the converse, let l(x - y) C band n [ N. Consider

ða _ lðn� xÞÞ ^ b

¼ ða _ a _ lðn� yþ y� xÞÞ ^ b

�ða _ ðb ^ lðn� yÞ ^ lðy� xÞÞÞ ^ b

� a _ ðb ^ lðn� yÞ ^ lðx� yÞÞ� a _ ðb ^ lðn� yÞÞ¼ ða _ bÞ ^ ða _ lðn� yÞÞ¼ b ^ ða _ lðn� yÞÞ¼ ða _ lðn� yÞÞ ^ b:

Similarly, we can prove that (a _ l(n - y)) ^ b C (a_ l(n - x)) ^ b. Hence (a _ l(n - y)) ^ b = (a _ l(n -

x)) ^ b. Thus xl = yl.

Corollary 4.4 If l is an equiprime fuzzy ideal of N, then

N/l is zerosymmetric nearring.

Proof Suppose that l is an equiprime fuzzy ideal of N.

By Theorem 4.2, N/l is a nearring. Let xl [ N/l. By

Lemma 3.9 (iii), l(x0) C b.

) lðx0� 0Þ� b

) x0 l ¼ 0l ðby Theorem 4:3Þ) xl0l ¼ 0l:

Thus N/l is zerosymmetric nearring.

Corollary 4.5 If l is a c-prime fuzzy ideal of N, then N/lis integral.

Proof Suppose that l is a c-prime fuzzy ideal of N. By

Theorem 4.2, N/l is a nearring. Let xl , yl [ N/l such that

xl�yl = 0l.

) xyl ¼ 0l

) lðxy� 0Þ� b:

As l is a c-prime fuzzy ideal of N, we get

a _ lðxÞ _ lðyÞ� b ^ lðxyÞ ¼ b:

) lðxÞ� b or lðyÞ� b

) xl ¼ 0l or yl ¼ 0l:

Hence N/l is integral.

Corollary 4.6 Let l be a fuzzy ideal of N. Then N/lb %N/l.

Proof Define /: N ? N/l by /(x) = xl. Then

/ðxþ yÞ ¼ xþyl ¼ xlþ yl ¼ /ðxÞ þ /ðyÞ;/ðxyÞ ¼ xyl ¼ xlyl ¼ /ðxÞ/ðyÞ:

Hence / is a homomorphism.

ker / ¼ fx 2 N j/ðxÞ ¼ /ð0Þg¼ fx 2 N j xl ¼ 0lg¼ fx 2 N j lðxÞ� bg ðby Theorem 4:3Þ¼ lb:

Thus N=lb ffi N=l:

Theorem 4.7 Let l be an equiprime fuzzy ideal of N.

Then N/l is an equiprime nearring.

Proof By Theorem 4.2, N/l is a nearring. We have to

prove that {0l} is an equiprime ideal of N/l. Let

ðalÞðrlÞðxlÞ � ðalÞðrlÞðylÞ ¼ 0l

for all rl [ N/l. Then arxl = aryl for all r [ N. By Theorem

4.3, we get l(arx - ary) C b for all r [ N. Now by Lemma

3.9(i), l(a) C b or l(x - y) C b. Again by Theorem 4.3,

we get al = 0l or x - yl = 0l. Thus N/l is an equiprime

nearring.

942 B. S. Kedukodi et al.

123

Theorem 4.8 Let l be a 3-prime (resp. c-prime) fuzzy

ideal of N. Then N/l is a 3-prime (resp. c-prime) nearring.

Theorem 4.9 Let f N ? M be an onto homomorphism

and l and r be fuzzy ideals of N and M respectively.

Suppose l has the sup property. Then

(i) N/l % M /f(l) ,

(ii) N/f-1(r) % M /r.

Proof (i) Denote f(l) = k. Define / N/l ? M/k by

/(xl) = f(x)k. Now,

xl ¼ yl, lðx� yÞ� b

, x� y 2 lb

, f ðx� yÞ 2 f ðlbÞ ¼ ðf ðlÞÞb ðby Remark 3:30Þ, f ðxÞ � f ðyÞ 2 ðf ðlÞÞb, f ðlÞðf ðxÞ � f ðyÞÞ� b

, f ðxÞk ¼ f ðyÞk

, /ðxlÞ ¼ /ðylÞ:

Hence / is well defined and one-one. It is easy to verify

that / is an onto homomorphism.

Thus N/l % M/f(l).

(ii) Denote f-1(r) = k. Define /: N/k? M/r by

/(xk) = f(x)r. Consider

xk ¼ yk, kðx� yÞ� b

, f�1ðrÞðx� yÞ� b

, rðf ðx� yÞÞ� b

, rðf ðxÞ � f ðyÞÞ� b

, f ðxÞr ¼ f ðyÞr

, /ðxkÞ ¼ /ðykÞ:

Hence / is well defined and one–one. It is easy to verify

that / is an onto homomorphism.

Thus N/ f-1(r) % M / r.

Theorem 4.10 Let l and k be fuzzy ideals of N. Then

k/l : N/l ? [0, 1] defined by (k/l)(xl) = k(x) is a fuzzy

ideal of N/l and

N=k ffi N=lk=l

:

Proof It is straightforward to prove that k/l is a fuzzy

ideal of N/l. Define f:N ? N/l by f(x) = xl. Then f is an

onto homomorphism. By Theorem 4.9 (ii), we get

N=f�1ðk=lÞ ffi N=lk=l

:

Now for x 2 N; f�1ðk=lÞðxÞ¼ ðk=lÞðf ðxÞÞ¼ ðk=lÞð xlÞ ¼ kðxÞ:

This completes the proof.

Let l be a fuzzy ideal of N. For brevity, if l an equi-

prime fuzzy ideal of N then we say l is e-prime. We denote

by Xlithe family of all i-prime (i [ {e, 3, c}) fuzzy ideals r

of N such that l � r; r has the same thresholds as that

of l and r(0) C l(0).

Definition 4.11 Let l be a fuzzy ideal of N. The i-prime

(i [ {e, 3, c}) fuzzy radical of l, denoted by <i, is defined

by <iðlÞ ¼Tfrjr 2 Xli

g:

Proposition 4.12 Let l be a fuzzy ideal of N and i [ {e,

3, c}. Then

(i) l � <iðlÞ:(ii) <i(l) = l if l is i-prime.

(iii) <i(l) is i-prime if l is i-prime.

(iv) <i(<i(l)) = <i(l) if l is i-prime.

(v) <i(l)(0) = l(0).

(vi) <i(l)(x) = l(0) for all x [ <i(l)*.

Proof Let i [ {e, 3, c} be fixed. The proofs of (i)–(iii) are

straightforward. (iv) follows from (ii) and (iii). To prove

(v), define a fuzzy subset r of N by r(x) = l(0) for all

x [ N. Then r [ Xl_i. Now <i(l)(0) B r(0) = l(0).

Also l(0) B <i(l)(0). Thus <i(l)(0) = l(0). The proof of

(vi) is immediate from (v).

Theorem 4.13 Let l be a fuzzy ideal of N and i [ {e, 3,

c}. Then <iðl�Þ � <iðlÞ�: If supflðxÞjx 2 Ng� lð0Þthen <i(l*) = <i(l)*.

Proof Let i [ {e, 3, c} be fixed. Let x [ <i(l*). Then x [ I

for all i-prime ideals I of N such that l� � I: Note that

Xli6¼/: Take r 2 Xli

: This implies x [ r* and hence r(x)

C r(0). Now

<iðlÞðxÞ ¼ ð\fr j r 2 XligÞðxÞ ¼ inffrðxÞ j r

2 Xlig� rð0Þ� lð0Þ:

This implies <i(l)(x) C l(0). By Proposition 4.12 (v), we

have l(0) = <i(l)(0). This implies <i(l)(x) C <i(l)(0).

Hence x [ <i(l)*. This proves <iðl�Þ � <iðlÞ�:

Now assume that supflðxÞjx 2 Ng� lð0Þ: Define a

fuzzy subset r by

rðyÞ ¼ lð0Þ if y 2 <iðl�Þ0 if y 62 <iðl�Þ

�

Then r is i-prime and l � r: Also as 0 [ <i(l*), we get

r(0) = l(0). Hence r [ Xl_i. Now let x [ <i(l)*. This

implies <i(l)(x) = <i(l)(0) = l(0) = r(0). But <i(l)(x) B

r(x). This implies r(0) B r(x). Hence x [ r* = <i(l*). This

proves <iðlÞ� � <iðl�Þ: Thus <i(l*) = <i(l)*.

Equiprime, 3-prime and c-prime fuzzy ideals of nearrings 943

123

Corollary 4.14 Let l be a non constant i - prime

(i [ {e, 3, c}) fuzzy ideal of N. Let supflðxÞjx 2 Ng� lð0Þ:If l* = {0} then N=<iðlÞ ffi N=PiðNÞ:

Proof Let i [ {e, 3, c} be fixed. By Theorem 4.10 of

Dutta and Biswas (1997), we have N/<i(l) % N/<i(l)*.

Now by Theorem 4.13, N/<i(l)* = N/<i(l*) = N/Pi(N)

(as l* = {0}).

Remark 4.15 Groenewald (1988, 1991) proved that Pc

and P3 are not Kurosh–Amitsur radicals in the class of all

nearrings. Booth et al. (1990) proved that Pe is a Kurosh-

Amitsur radical in a class of nearrings. Also, if N is an

equiprime nearring then Pe(N) = {0}. Equiprime nearrings

are extensively studied by Veldsman (1992).

Theorem 4.16 Let l be a fuzzy ideal of N. Then

(i) <3ðlÞ � <eðlÞ:(ii) if N is a ring then <3(l) = <e(l).

(iii) <3ðlÞ � <cðlÞ:(iv) if N is a commutative ring then

<3(l) = <c(l) = <e(l).

Proof The proof is clear from Theorems 3.13, 3.15 and

3.16.

5 Conclusions

In this paper, we have introduced the notions of equiprime

fuzzy ideal, 3-prime fuzzy ideal and c-prime fuzzy ideal of

a nearring using the concept of thresholds. The concept of

thresholds empowers the examples to exhibit the fuzzy

character. We have also characterized the above three types

of fuzzy ideals in terms of level subsets and fuzzy points.

We have defined fuzzy cosets based on generalized fuzzy

ideals and proved fundamental results.

Acknowledgments The authors thank Prof. Stefan Veldsman,

Sultan Qaboos University, Sultanate of Oman, for his comments and

suggestions. The authors also thank the anonymous referees for their

constructive comments. The first and the second author acknowledge

Manipal University and the third author acknowledges Acharya

Nagarjuna University for their encouragement.

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