New types of fuzzy ideals of near-rings - Semantic Scholar
Neural Comput & Applic (2012) 21:863–868 DOI 10.1007/s00521-011-0570-1
ORIGINAL ARTICLE
New types of fuzzy ideals of near-rings Jianming Zhan • Yunqiang Yin
Received: 24 May 2010 / Accepted: 11 March 2011 / Published online: 5 April 2011 Ó Springer-Verlag London Limited 2011
Abstract In this paper, we consider the ð2c ; 2c _ qd Þfuzzy and ð2c ; 2c _ qd Þ-fuzzy subnear-rings (ideals) of a near-ring. Some new characterizations are also given. In particular, we introduce the concepts of (strong) prime ð2c ; 2c _ qd Þ-fuzzy ideals of near-rings and discuss the relationship between strong prime ð2c ; 2c _ qd Þ-fuzzy ideals and prime ð2c ; 2c _ qd Þ-fuzzy ideals of near-rings. Keywords Near-ring Subnear-ring (ideal) Prime ideal Prime ð2c ; 2c _ qd Þ-fuzzy subnear-ring (ideal) ð2c ; 2c _ qd Þ-fuzzy subnear-ring (ideal) 1 Introduction A near-ring satisfying all axioms of an associative ring, expect for commutativity of addition and one of the two distributive laws. Abou-Zaid [1] introduced the concepts of fuzzy subnear-rings (ideals) and studied some of their related properties in near-rings. The concept was discussed further by many researchers, for example, see [3–7, 11, 13, 14]. A new type of fuzzy subgroup, that is, the ð2; 2 _ qÞfuzzy subgroup, was introduced in an earlier paper of Bhakat and Das [2] using the combined notions of ‘‘belongingness’’ and ‘‘quasicoincidence’’ of fuzzy points and fuzzy sets. In fact, the ð2; 2 _ qÞ-fuzzy subgroup is an J. Zhan (&) Department of Mathematics, Hubei University for Nationalities, Enshi 445000, China e-mail: [email protected] Y. Yin School of Mathematics and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, China e-mail: [email protected]
important generalization of Rosenfeld’s fuzzy subgroup. It is now natural to investigate similar type of generalizations of the existing fuzzy subsystems with other algebraic structures, see [3, 4, 9–12, 14]. In [3], Davvaz introduced the concepts of ð2; 2 _ qÞfuzzy subnear-rings (ideals) of near-rings and investigated some of their related properties. Further, one of present author [11] considered the concept of ð2; 2 _ qÞ-fuzzy subnear-rings (ideals) of near-rings and obtained some of its related properties. Finally, some characterizations of [l]t by means of ð2; 2 _ qÞ-fuzzy ideals were also given. In particular, we [14] redefined generalized fuzzy subnearrings (ideals) of a near-ring and investigated some of their related properties. Further, we discussed the relationship between strong prime ð2; 2 _ qÞ-fuzzy ideals and prime ð2; 2 _ qÞ-fuzzy ideals in near-rings. In this paper, we consider the ð2c ; 2c _ qd Þ-fuzzy and ð2c ; 2c _ qd Þ-fuzzy subnear-rings (ideals) of a near-ring. In particular, we introduce the concept of strong prime ð2c ; 2c _ qd Þ-fuzzy ideals of near-rings and discuss the relationship between strong prime ð2c ; 2c _ qd Þ-fuzzy ideals and prime ð2c ; 2c _ qd Þ-fuzzy ideals of near-rings.
2 Preliminaries A non-empty set R with two binary operation ‘‘?’’ and ‘‘’’ is called a left near-ring if it satisfies the following conditions: (1) (2) (3)
(R, ?) is a group, ðR; Þ is a semigroup, x ðy þ zÞ ¼ x y þ x z, for all x; y; z 2 R.
We will use the word ‘‘near-ring’’ to mean ‘‘ left nearring’’ and denote xy instead of x y.
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An ideal I of a near-ring R is a subset of R such that (i) (I, ?) is a normal subgroup of (R, ?), (ii) RI I, (iii) ðx þ aÞy xy 2 I for any a 2 I and x; y 2 R. An ideal P of R is called prime if IJ P implies I P or J P for all ideals I and J of R. In what follows, let R be a near-ring unless otherwise specified. A fuzzy set l of R of the form tð6¼ 0Þ if y ¼ x; lðyÞ ¼ 0 if y 6¼ x; is said to be a fuzzy point with support x and value t and is denoted by xt. A fuzzy point xt is said to belong to (resp., be quasi-coincident with) a fuzzy set l, written as xt 2 l (resp., xt q l) if l(x) C t (resp., l(x) ? t [ 1). If xt 2 l or xt q l, then, we write xt 2 _ ql. If l(x) \ t (resp., l(x) ? t B 1), then, we call xt 2 l (resp., xt q l). We note that the symbol 2 _ q means that 2 _ q does not hold. Now, we give the concept of the product of two fuzzy sets of R. Definition 2.1 Let l and m be fuzzy sets of R. Then, the product of l and m is defined by _ ðl mÞðxÞ ¼ ðlðaÞ ^ mðbÞÞ
Theorem 3.1 A fuzzy set l of R is an ð2c ; 2c _ qd Þ-fuzzy subnear-ring of R if and only if for any x; y; a 2 R, (F2a) l(x ? y) _ c C l(x) ^ l(y) ^ d, (F2a’) l(- x) _ c C l(x) ^ d, (F2b) l(xy) _ c C l(x) ^ l (y) ^ d. Moreover, l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R if l satisfies the above conditions and (F2c) l(y ? x - y) _ c C l(x) ^ d, (F2d) l(xy) _ c C l(y) ^ d, (F3e) l((x ? a)y - xy) _ c C l(a) ^ d. Proof We only prove (F1a),(F2a). The others are similar. (F1a))(F2a) If there exist x; y 2 R such that l(x ? y) _ c \ r = l(x) ^ l(y) ^ d, then l(x) C r [ c, l(y) C r [ c, l(x ? y) \ r and l(x ? y) ? r \ 2r B 2d, that is, xr 2c l; yr 2c , but ðx þ yÞr 2c _ qd l, a contradiction. Hence (F2a) holds. (F2a))(F1a) If there exist x; y 2 R and t; r 2 ðc; 1 such that xt 2c l; yr 2c l, but ðx þ yÞt ^ r 2c _ qd l, then, l(x) C t, l(y) C r, l(x ? y) \ t ^ r and l(x ? y) ? t ^ r B 2d. It follows that l(x ? y) \ d, and so l(x ? y) _ c \ t ^ r ^ d B l(x) ^ l (y) ^ d, a contradiction. h
x¼ab
and ðl mÞ (x) = 0 if x cannot be expressed as x = ab.
Remark 3.1 For any ð2c ; 2c _ qd Þ-fuzzy ideal l of R, we can conclude that
3 ð2c ; 2c _ qd Þ-Fuzzy ideals
(1)
In this Section, we introduce the concept of ð2c ; 2c _ qd Þfuzzy ideals of near-rings and investigate some new results. Let c; d 2 ½0; 1 be such that c \ d. For a fuzzy point xr and a fuzzy set l of R, we say that
(2)
(1) (2) (3)
xr 2c l if l(x) C r [ c. xr qd l if l(x) ? r [ 2d. xr 2c _ qd l if xr 2c l or xr qd l.
Definition 3.1 A fuzzy set l of R is called an ð2c ; 2c _ qd Þ-fuzzy subnear-ring of R if for all t; r 2 ðc; 1 and x; y; a 2 R, (F1a) xt 2c l and yr 2c l imply ðx þ yÞt ^ r 2c _ qd l, (F1a’) xt 2c l implies ðxÞt 2c _ qd l, (F1b) xt 2c l and yr 2c l imply ðxyÞt ^ r 2c _ qd l. Moreover, l is called an ð2c ; 2c _ qd Þ-fuzzy ideal of R if l is ð2c ; 2c _ qd Þ-fuzzy subnear-ring of R and (F1c) xr 2c l implies ðy þ x yÞr 2c _ qd l, (F1d) yr 2c l and x 2 R imply ðxyÞr 2c _ qd l, (F1e) ar 2c l implies ððx þ aÞy xyÞr 2c _ qd l.
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(3) (4)
if c = 0 and d = 1, then, l is the fuzzy ideal of R (see [1]); if c = 0 and d = 0.5, then l is the ð2; 2 _ qÞ-fuzzy ideal of R (see [3]); if c = 0.5 and d = 1, then l is the ð2; 2 _ qÞ-fuzzy ideal of R (see [11]); l is the fuzzy ideal of R with thresholds (c, d) (see [3]).
For any fuzzy set l of R, we define lcr ¼ fx 2 Rjxr 2c lg, ldr ¼ fx 2 Rjxr qd lg, and ½ldr ¼ fx 2 Rjxr 2c _ qd lg for all r 2 ½0; 1. It is clear that [l]dr = lcr [ ldr . The next theorem provides the relationship between ð2c ; 2c _ qd Þ-fuzzy ideals of R and crisp ideals of R. Theorem 3.2 (1) (2)
(3)
Let l be a fuzzy set of R. Then,
l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R if and only if lcr (=[) is an ideal of R for all r 2 ðc; d. If 2d = 1 ? c, then, l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R if and only if ldr (=[) is an ideal of R for all r 2 ðd; 1. If 2d = 1 ? c, then, l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R if and only if [l]dr (=[) is an ideal of R for all r 2 ðc; 1.
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Proof (1)
(2) (3)
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(2)
Let l be an ð2c ; 2c _ qd Þ-fuzzy ideal of R and x; y 2 lcr for all r 2 ðc; d, then, l(x) C r [ c and l(y) C r [ c. It follows that l(x ? y) _ c C l(x) ^ l (y) ^ d C r ^ d = r, that is, x þ y 2 lcr . Similarly, we can prove the other conditions of ideals hold. Hence, lcr is an ideal of R for all r 2 ðc; d. Conversely, assume that lcr is an ideal of R for all r 2 ðc; d. Let x; y 2 R. If l(x ? y) _ c \ r = l(x) ^ l (y) ^ d, then, xr 2c l; yr 2c l, but ðx þ yÞr 2c _ qd l, that is, x; y 2 lcr . Since lcr is an ideal, we have x þ y 2 lcr , a contradiction. Hence (F2a) holds. Similarly, we can prove (F2a’), (F2b), (F2c), (F2d), and (F2e) hold. Thus, l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R. The proof is similar to the proof of (1). Let l be an ð2c ; 2c _ qd Þ-fuzzy ideal of R and r 2 ðc; 1. Then, for all x; y 2 ½ldr , we have xr 2c _ qd l; yr 2c _ qd l, that is, l(x) C r [ c or l(x) [ 2d - r [ 2d - 1 = c, and l(y) C r [ c or l(y) [ 2d - r [ 2d - 1 = c. Since l is an ð2c ; 2c _ qd Þfuzzy ideal of R, then l(x ? y) _ c C l(x) ^ l (y) ^ d, and so l(x ? y) C l(x) ^ l (y) ^ d.
Case 1 r 2 ðc; d. Then, 2d - r C d C r and so l(x ? y) C r ^ r ^ d = r or l(x ? y) C r ^ (2d - r) ^ d = r or l(x ? y) C (2d - r) ^ (2d - r) ^ d = d C r. Hence ðx þ yÞr 2c l: Case 2 r 2 ðd; 1. Then, 2d - r \ d \ r and so l(x ? y) C r ^ r ^ d = d [ 2d - r or l(x ? y) [ r ^ (2d - r) ^ d = 2d - r or l(x ? y) [ (2d - r) ^ (2d - r) ^ d = 2d - r. Hence (x ? y)r qdl. Thus, in any case, ðx þ yÞr 2c _ qd l, i.e., x þ y 2 ½ldr . Similarly, we can prove the other conditions. Hence [l]dr is an ideal of R. Conversely, assume that [l]dr is an ideal of R for all r 2 ðc; d. Let x; y 2 R. If l(x ? y) _ c \ r = l(x) ^ l (y) ^ d, then xr 2c l; yr 2c l; but ðx þ yÞr 2c _ qd l;
(3)
Definition 3.2 A fuzzy set l of R is called an ð2c ; 2c _ qd Þ-fuzzy subnear-ring of R if for all t; r 2 ðc; 1 and x; y; a 2 R, (F3a) ðx þ yÞt ^ r 2c l implies xt 2c _ qd l or yr 2c _ qd l, (F3a’) ðxÞt 2c l implies xt 2c _ qd l, (F3b) ðxyÞt ^ r 2c implies xt 2c _ qd l or yr 2c _ qd l. Moreover, l is called an ð2c ; 2c _ qd Þ-fuzzy ideal of R if l is an ð2c ; 2c _ qd Þ-fuzzy subnear-ring of R and (F3c) ðy þ x yÞr 2c l implies xt 2c _ qd l, (F3d) ðxyÞr 2c l and x 2 R imply yr 2c _ qd l, (F3e) ððx þ aÞy xyÞr 2c l implies ar 2c _ qd l. Theorem 3.3 A fuzzy set l of R is an ð2c ; 2c _ qd Þ-fuzzy subnear-ring of R if and only if for any x; y; a 2 R, (F4a) l(x ? y) _ d C l(x) ^ l(y), (F4a’) l(- x) _ d C l(x), (F4b) l(xy) _ d C l(x) ^ l (y). Moreover, l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R if l satisfies the above conditions and (F4c) l(y ? x - y) _ d C l(x), (F4d) l(xy) _ d C l(y), (F4e) l((x ? a)y - xy) _ d C l(a). Proof
If we take c = 0 and d = 0.5 in Theorem 3.2, we can conclude the following result: Corollary 3.1 (1)
Let l be a fuzzy set of R. Then,
l is an ð2; 2 _ qÞ-fuzzy ideal of R if and only if lr(=[) is an ideal of R for all r 2 ð0; 0:5 (see [3]).
The proof is similar to the proof of Theorem 3.1. h
Remark 3.2 For any ð2c ; 2c _ qd Þ-fuzzy ideal l of R, we can conclude that if d = 0.5, then l is the ð2; 2 _ qÞfuzzy ideal of R (see [7]). Theorem 3.4 (1) (2)
that is, x; y 2 ½ldr . Since lcr is an ideal, we have x þ y 2 ½ldr , a contradiction. Hence (F2a) holds. Similarly, we can prove (F2a’), (F2b), (F2c), (F2d), and (F2e) hold. Thus, l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R. h
l is an ð2; 2 _ qÞ-fuzzy ideal of R if and only if Q(l;r)(=[) is an ideal of R for all r 2 ð0:5; 1, where Qðl; rÞ ¼ fx 2 Rjxr q lg. l is an ð2; 2 _ qÞ-fuzzy ideal of R if and only if [l]r(=[) is an ideal of R for all r 2 ð0; 1 (see [7]).
l if l if
Proof
Let l be a fuzzy set of R. Then
is an ð2c ; 2c _ qd Þ-fuzzy ideal of R if and only lcr (=[) is an ideal of R for all r 2 ðd; 1. is an ð2c ; 2c _ qd Þ-fuzzy ideal of R if and only ldr (=[) is an ideal of R for all r 2 ðc; d. The proof is similar to the proof of Theorem 3.2. h
If we take d = 0.5 in Theorem 3.4, we can conclude the following result: Corollary 3.2 (1)
Let l be a fuzzy set of R. Then
l is an ð2; 2 _ qÞ-fuzzy ideal of R if and only if lr(=[) is an ideal of R for all r 2 ð0:5; 1 (see [7]).
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l is an ð2; 2 _ qÞ-fuzzy ideal of R if and only if Q(l;r)(=[) is an ideal of R for all r 2 ð0; 0:5.
If we take c = 0 and d = 0.5 in Theorem 4.2, we can conclude the following result:
4 (Strong) prime ð2c ; 2c _ qd Þ-fuzzy ideals In this Section, we introduce the concepts of prime and strong prime ð2c ; 2c _ qd Þ-fuzzy ideals of near-rings. In particular, we discuss the relationship between strong prime ð2c ; 2c _ qd Þ-fuzzy ideals and prime ð2c ; 2c _ qd Þfuzzy ideals of near-rings. Definition 4.1 An ð2c ; 2c _ qd Þ-fuzzy ideal l of R is called prime if for all x; y 2 R and t 2 ðc; 1, we have (P) ðxyÞt 2c l ) xt 2c _ qd l or yt 2c _ qd l: Theorem 4.1 An ð2c ; 2c _ qd Þ-fuzzy ideal l of R is prime if for all x; y 2 R, it satisfies: (P’) l(x) _ l(y) _ c C l(xy) ^ d. Proof Let l be a prime ð2c ; 2c _ qd Þ-fuzzy ideal of R. If there exist x; y 2 R such that l(x) _ l(y) _ c \ t = l(xy) ^ d, then c\t d; ðxyÞt 2c l, but xt 2c l and yt 2c l. Since l(x) ? t \ 2t B 2d and l(y) ? t \ 2t B 2d, then xt qd l and yt qd l, and hence, we have xt 2c _ qd l and yt 2c _ qd l, which is a contradiction. Thus, (P’) holds. Conversely, suppose that condition (P’) holds. Let ðxyÞt 2c l. Then, l(xy) C t and so l(x) _ l(y) C l(xy) ^ d C t ^ d. We consider the following two cases: If t 2 ðc; d, then l(x) C t or l(y) C t, that is, xt 2c l or yt 2c l. Thus, xt 2c _ qd l or yt 2c _ qd l. (ii) If t 2 ðd; 1, then l(x) _ l(y) C d, and so, l(x) C d or l(y) C d. Hence, l(x) ? t [ 2d or l(y) ? t [ 2d, that is, xtqdl or ytqdl. Thus, xt 2c _ qd l or yt 2c _ qd l. (i)
This proves that l is prime.
h
Theorem 4.2 An ð2c ; 2c _ qd Þ-fuzzy ideal l of R is prime if and only if lct (=[) is a prime ideal of R for all t 2 ðc; d. Proof Let l be a prime ð2c ; 2c _ qd Þ-fuzzy ideal of R and t 2 ðc; d. Then, by Theorem 3.2, lct is an ideal of R for all c \ t B d. Let xy 2 lct . By Theorem 4.1, we have l(x) _ l(y) C l(xy) ^ d C t ^ d = t, and so l(x) C t or l(y) C t. Thus, x 2 lct or y 2 lct . This shows that lct is a prime ideal of R, for all t 2 ðc; d. Conversely, assume that lct (=[) is a prime ideal of R for all t 2 ðc; d. Then by Theorem 3.2, l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R. Let ðxyÞt 2c l. Then xy 2 lct . Since lct is prime, x 2 lct or y 2 lct , that is, xt 2
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l or yt 2 l. Thus, xt 2c _ qd l or yt 2c _ qd l. Therefore, l must be a prime ð2c ; 2c _ qd Þ-fuzzy ideal of R. h
Corollary 4.1 Let l be a fuzzy set of R. Then l is a prime ð2; 2 _ qÞ-fuzzy ideal of R if and only if lt(=[) is a prime ideal of R for all t 2 ð0; 0:5 (see [7]). Theorem 4.3 If 2d = 1 ? c, then a fuzzy set l of R is a prime ð2c ; 2c _ qd Þ-fuzzy ideal if and only if ldr (=[) is a prime ideal of R for all t 2 ðd; 1. Proof Let l be a prime ð2c ; 2c _ qd Þ-fuzzy ideal of R and t 2 ðd; 1. Then by Theorem 3.2(2), ldt is an ideal of R and t [ d [ 2d - t. To prove ldr is prime, let xy 2 ldt . Hence l(xy) [ 2d - t. Since l is a prime ð2c ; 2c _ qd Þfuzzy ideal of R, we have l(x) _ l(y) _ c C l(xy) ^ d [ 2d - t C 2d - 1 = c. It follows that l(x) _ l(y) C 2d - t, and so x 2 ldt or y 2 ldt : Therefore, ldr is a prime ideal of R. Conversely, let ldr (=[) be a prime ideal of R for all t 2 ðc; 1, then by Theorem 3.2(2), we know l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R. Now, if there exist x; y 2 R such that l(x) _ l(y) _ c \ t = l(xy) ^ d. Then, t B d, l(xy) C t, l(x) \ t, l(x) ? t \ 2t B 2d, l(y) \ t, and l(y) ? t \ 2t B 2d. It follows that ðxyÞt 2c l, but xt 2c _ qd l and yt 2c _ qd l, a contradiction. Therefore, l(x) _ l(y) _ c C l(xy) ^ d for all x; y 2 R and so l is a prime ð2c ; 2c _ qd Þ-fuzzy ideal of R. h Theorem 4.4 If 2d = 1 ? c, then a fuzzy set l of R is a prime ð2c ; 2c _ qd Þ-fuzzy ideal if and only if [l]dt (=[) is a prime ideal of R for all t 2 ðc; 1. Proof Let l be a prime ð2c ; 2c _ qd Þ-fuzzy ideal of R. Then by Theorem 3.2(3), [l]dt is an ideal of R for all t 2 ðc; 1. To prove [l]dt is prime, let xy 2 ½ldt . Since [l]dt = h. ldt [ lct , we have xy 2 ldt or xy 2 lct : Case 1 xy 2 ldt lct . Then, l(xy) ? t [ 2d and l(xy) \ t. (1)
(2)
If l(xy) B d, then l(x) _ l(y) ? t C l(xy) ^ d ? t = l(xy) ? t [ 2d, which implies, l(x) ? t [ 2d or l(y) ? t [ 2d, that is, x 2 ldt ½ldt or y 2 ldt ½ldt . If l(xy) [ d, then d \ l(xy) \ t. Thus, l(x) _ l(y) ? t C l(xy) ^ d ? t = d ? t [ 2d. Hence x 2 ldt ½ldt or y 2 ldt ½ldt :
Case 2 (1)
xy 2 lct . Then, l(xy) C t.
If t B d, then l(x) _ l(y) C l(xy) ^ d C t, which implies, x 2 lct ½ldt or y 2 lct ½ldt :
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If t [ d, then l(x) _ l(y) C t ^ d = d, which implies, l(x) _ l(y) ? t [ 2d. Hence, x 2 ldt ½ldt or y 2 ldt ½ldt :
Therefore, [l]dt is a prime ideal of R. Conversely, let [l]dt (=[) be a prime ideal of R for all t 2 ðc; 1, then by Theorem 3.2(3), we know l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R. Let ðxyÞt 2c l, then xy 2 lct ½ldt . Since [l]dt is prime, we have x 2 ½ldt or y 2 ½ldt . This implies, xt 2c _ qd l or yt 2c _ qd l. Thereh fore, l is a prime ð2c ; 2c _ qd Þ-fuzzy ideal of R. If we take c = 0 and d = 0.5 in Theorem 4.4, we can conclude the following result: Corollary 4.2 Let l be a fuzzy set of R. Then, l is a prime ð2; 2 _ qÞ-fuzzy ideal of R if and only if [lt](=[) is a prime ideal of R for all t 2 ð0; 1 (see [7]). Now, we give the concept of strong prime ð2c ; 2c _ qd Þfuzzy ideals of near-rings. Definition 4.2 An ð2c ; 2c _ qd Þ-fuzzy ideal q of R is called strong prime if for every ð2c ; 2c _ qd Þ-fuzzy ideals l and m of R, it satisfies: (P’’) l m q implies l q or m q: Example 4.1 Let R = {0, a, b, c} be Klein’s four group. Define multiplication in N as follows:
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W l(x) C t. Hence, ðtI tJ ÞðxÞ ¼ x¼yz tI ðyÞ ^ tJ ðzÞ t lðxÞ. Therefore, tI tJ l. Since l is a strong prime ð2c ; 2c _ qd Þ-fuzzy ideal of R, we have tI l or tJ l, this implies I lct or J lct . This completes the proof. h The following is a consequence of Theorems 4.2 and 4.5. Theorem 4.6 Every strong prime ð2c ; 2c _ qd Þ-fuzzy ideal of a near-ring is a prime ð2c ; 2c _ qd Þ-fuzzy ideal. If we take c = 0 and d = 0.5 in Theorem 4.6, we can conclude the following result: Corollary 4.3 Every strong prime ð2; 2 _ qÞ-fuzzy ideal of a near-ring is a prime ð2; 2 _ qÞ-fuzzy ideal (see [10]). Remark 4.1 The converse of Theorem 4.6 is not true in general as shown in the following example. Example 4.2 Let ðZ; þ; Þ be the near-ring (it is also a ring) of all integers. Define fuzzy sets l, m and x of Z as follows: 8x 2 Z 8 x ¼ 0; < 1 lðxÞ ¼ 0:4 x 2 ð2Þnf0g; : otherwise 80 x ¼ 0; < 1 mðxÞ ¼ 0:4 x 2 ð3Þnf0g and : otherwise 80 x ¼ 0; < 1 xðxÞ ¼ 0:5 x 2 ð4Þnf0g; : 0 otherwise Then, l, m, and x are ð2c ; 2c _ qd Þ-fuzzy ideal of Z and non-empty subset lct is a prime ideal of R for all t 2 ðc; d. By Theorem 4.2, we know that l is a prime ð2c ; 2c _ qd Þfuzzy ideal of Z, but it is not strong prime. In fact, m x l, but m * l and x * l.
Then ðR; þ; Þ is a near-ring [8]. Define a fuzzy set l of R as follows: lð0Þ ¼ 0:7; lðaÞ ¼ 0:7; lðbÞ ¼ 0:4 and lðcÞ ¼ 0:3: Then l is a strong prime ð20:4 ; 20:4 _q0:7 Þ-fuzzy ideal of R. Theorem 4.5 Let l be a strong prime ð2c ; 2c _ qd Þ-fuzzy ideal of R. Then lct (=[) is a prime ideal of R for all t 2 ðc; d. Proof Let t 2 ðc; d be such that lct is non-empty. Then lct is an ideal of R by Theorem 3.2(1). Now we show that lct is prime. Let I and J be two ideals of R such that IJ lct . Then it is easy to see that tI and tJ are two ð2c ; 2c _ qd Þfuzzy ideals of R and that tI tJ l. In fact, let x 2 R. If (tI tJ)(x) = 0, then (tI tJ)(x) = 0 B l(x). Otherwise, there exist a; b 2 R such that x = ab and tI(a) ^ tJ(b) = 0. This implies a 2 I and b 2 J, hence x 2 IJ lct , that is,
Acknowledgments Supported by the National Natural Science Foundation of China (60875034); the Innovation Term of Education Committee of Hubei Province, China (T201109) and the Natural Science Foundation of Hubei Province, China (2009CDB340).
References 1. Abou-Zaid A (1996) On fuzzy subnear-rings. Fuzzy Sets Syst 81:383–393 2. Bhakat SK, Das P (1996) ð2; 2 _qÞ-fuzzy subgroups. Fuzzy Sets Syst 80:359–368 3. Davvaz B (2006) ð2; 2 _ qÞ-fuzzy subnear-rings and ideals. Soft Comput 10:206–211 4. Davvaz B (2008) Fuzzy R-subgroups with thresholds of nearrings and implication operators. Soft Comput 12:875–879 5. Hong SM, Jun YJ, Kim HS (1998) Fuzzy ideals in near-rings. Bull Koran Math Soc 35:343–348 6. Kedukodi BS, Kuncham SP, Bhavanari S (2009) Equiprime, 3-prime and c-prime fuzzy ideals of nearrings. Soft Comput 13:933–944
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868 7. Kim SD, Kim HS (1996) On fuzzy ideals of near-rings. Bull Koran Math Soc 33:593–601 8. Pilz G (1983) Near-Rings, 2nd edn, North-HollandMathematics studies, vol 23. North-Holland, Amsterdam 9. Yin Y, Zhan J (2010) New types of fuzzy filters of BL-algebras. Comput Math Appl 60:2115–2125 10. Yuan X, Zhang C, Ren Y (2003) Generalized fuzzy groups and many valued applications. Fuzzy Sets Syst 38:205–211
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Neural Comput & Applic (2012) 21:863–868 11. Zhan J, Davvaz B (2009) Generalized fuzzy ideals of near-rings. Appl Math J Chin Univ Ser B 24:343–349 12. Zhan J, Dudek WA (2007) Fuzzy h-ideals of hemirings. Inform Sci 177:876–886 13. Zhan J, Ma X (2005) Intuitionistic fuzzy ideals of near-rings. Sci Math Japon 61:219–223 14. Zhan J, Yin Y (2010) Redefined generalized fuzzy ideals of nearrings. Appl Math J Chin Univ Ser B 25(3):341–348
ORIGINAL ARTICLE
New types of fuzzy ideals of near-rings Jianming Zhan • Yunqiang Yin
Received: 24 May 2010 / Accepted: 11 March 2011 / Published online: 5 April 2011 Ó Springer-Verlag London Limited 2011
Abstract In this paper, we consider the ð2c ; 2c _ qd Þfuzzy and ð2c ; 2c _ qd Þ-fuzzy subnear-rings (ideals) of a near-ring. Some new characterizations are also given. In particular, we introduce the concepts of (strong) prime ð2c ; 2c _ qd Þ-fuzzy ideals of near-rings and discuss the relationship between strong prime ð2c ; 2c _ qd Þ-fuzzy ideals and prime ð2c ; 2c _ qd Þ-fuzzy ideals of near-rings. Keywords Near-ring Subnear-ring (ideal) Prime ideal Prime ð2c ; 2c _ qd Þ-fuzzy subnear-ring (ideal) ð2c ; 2c _ qd Þ-fuzzy subnear-ring (ideal) 1 Introduction A near-ring satisfying all axioms of an associative ring, expect for commutativity of addition and one of the two distributive laws. Abou-Zaid [1] introduced the concepts of fuzzy subnear-rings (ideals) and studied some of their related properties in near-rings. The concept was discussed further by many researchers, for example, see [3–7, 11, 13, 14]. A new type of fuzzy subgroup, that is, the ð2; 2 _ qÞfuzzy subgroup, was introduced in an earlier paper of Bhakat and Das [2] using the combined notions of ‘‘belongingness’’ and ‘‘quasicoincidence’’ of fuzzy points and fuzzy sets. In fact, the ð2; 2 _ qÞ-fuzzy subgroup is an J. Zhan (&) Department of Mathematics, Hubei University for Nationalities, Enshi 445000, China e-mail: [email protected] Y. Yin School of Mathematics and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, China e-mail: [email protected]
important generalization of Rosenfeld’s fuzzy subgroup. It is now natural to investigate similar type of generalizations of the existing fuzzy subsystems with other algebraic structures, see [3, 4, 9–12, 14]. In [3], Davvaz introduced the concepts of ð2; 2 _ qÞfuzzy subnear-rings (ideals) of near-rings and investigated some of their related properties. Further, one of present author [11] considered the concept of ð2; 2 _ qÞ-fuzzy subnear-rings (ideals) of near-rings and obtained some of its related properties. Finally, some characterizations of [l]t by means of ð2; 2 _ qÞ-fuzzy ideals were also given. In particular, we [14] redefined generalized fuzzy subnearrings (ideals) of a near-ring and investigated some of their related properties. Further, we discussed the relationship between strong prime ð2; 2 _ qÞ-fuzzy ideals and prime ð2; 2 _ qÞ-fuzzy ideals in near-rings. In this paper, we consider the ð2c ; 2c _ qd Þ-fuzzy and ð2c ; 2c _ qd Þ-fuzzy subnear-rings (ideals) of a near-ring. In particular, we introduce the concept of strong prime ð2c ; 2c _ qd Þ-fuzzy ideals of near-rings and discuss the relationship between strong prime ð2c ; 2c _ qd Þ-fuzzy ideals and prime ð2c ; 2c _ qd Þ-fuzzy ideals of near-rings.
2 Preliminaries A non-empty set R with two binary operation ‘‘?’’ and ‘‘’’ is called a left near-ring if it satisfies the following conditions: (1) (2) (3)
(R, ?) is a group, ðR; Þ is a semigroup, x ðy þ zÞ ¼ x y þ x z, for all x; y; z 2 R.
We will use the word ‘‘near-ring’’ to mean ‘‘ left nearring’’ and denote xy instead of x y.
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An ideal I of a near-ring R is a subset of R such that (i) (I, ?) is a normal subgroup of (R, ?), (ii) RI I, (iii) ðx þ aÞy xy 2 I for any a 2 I and x; y 2 R. An ideal P of R is called prime if IJ P implies I P or J P for all ideals I and J of R. In what follows, let R be a near-ring unless otherwise specified. A fuzzy set l of R of the form tð6¼ 0Þ if y ¼ x; lðyÞ ¼ 0 if y 6¼ x; is said to be a fuzzy point with support x and value t and is denoted by xt. A fuzzy point xt is said to belong to (resp., be quasi-coincident with) a fuzzy set l, written as xt 2 l (resp., xt q l) if l(x) C t (resp., l(x) ? t [ 1). If xt 2 l or xt q l, then, we write xt 2 _ ql. If l(x) \ t (resp., l(x) ? t B 1), then, we call xt 2 l (resp., xt q l). We note that the symbol 2 _ q means that 2 _ q does not hold. Now, we give the concept of the product of two fuzzy sets of R. Definition 2.1 Let l and m be fuzzy sets of R. Then, the product of l and m is defined by _ ðl mÞðxÞ ¼ ðlðaÞ ^ mðbÞÞ
Theorem 3.1 A fuzzy set l of R is an ð2c ; 2c _ qd Þ-fuzzy subnear-ring of R if and only if for any x; y; a 2 R, (F2a) l(x ? y) _ c C l(x) ^ l(y) ^ d, (F2a’) l(- x) _ c C l(x) ^ d, (F2b) l(xy) _ c C l(x) ^ l (y) ^ d. Moreover, l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R if l satisfies the above conditions and (F2c) l(y ? x - y) _ c C l(x) ^ d, (F2d) l(xy) _ c C l(y) ^ d, (F3e) l((x ? a)y - xy) _ c C l(a) ^ d. Proof We only prove (F1a),(F2a). The others are similar. (F1a))(F2a) If there exist x; y 2 R such that l(x ? y) _ c \ r = l(x) ^ l(y) ^ d, then l(x) C r [ c, l(y) C r [ c, l(x ? y) \ r and l(x ? y) ? r \ 2r B 2d, that is, xr 2c l; yr 2c , but ðx þ yÞr 2c _ qd l, a contradiction. Hence (F2a) holds. (F2a))(F1a) If there exist x; y 2 R and t; r 2 ðc; 1 such that xt 2c l; yr 2c l, but ðx þ yÞt ^ r 2c _ qd l, then, l(x) C t, l(y) C r, l(x ? y) \ t ^ r and l(x ? y) ? t ^ r B 2d. It follows that l(x ? y) \ d, and so l(x ? y) _ c \ t ^ r ^ d B l(x) ^ l (y) ^ d, a contradiction. h
x¼ab
and ðl mÞ (x) = 0 if x cannot be expressed as x = ab.
Remark 3.1 For any ð2c ; 2c _ qd Þ-fuzzy ideal l of R, we can conclude that
3 ð2c ; 2c _ qd Þ-Fuzzy ideals
(1)
In this Section, we introduce the concept of ð2c ; 2c _ qd Þfuzzy ideals of near-rings and investigate some new results. Let c; d 2 ½0; 1 be such that c \ d. For a fuzzy point xr and a fuzzy set l of R, we say that
(2)
(1) (2) (3)
xr 2c l if l(x) C r [ c. xr qd l if l(x) ? r [ 2d. xr 2c _ qd l if xr 2c l or xr qd l.
Definition 3.1 A fuzzy set l of R is called an ð2c ; 2c _ qd Þ-fuzzy subnear-ring of R if for all t; r 2 ðc; 1 and x; y; a 2 R, (F1a) xt 2c l and yr 2c l imply ðx þ yÞt ^ r 2c _ qd l, (F1a’) xt 2c l implies ðxÞt 2c _ qd l, (F1b) xt 2c l and yr 2c l imply ðxyÞt ^ r 2c _ qd l. Moreover, l is called an ð2c ; 2c _ qd Þ-fuzzy ideal of R if l is ð2c ; 2c _ qd Þ-fuzzy subnear-ring of R and (F1c) xr 2c l implies ðy þ x yÞr 2c _ qd l, (F1d) yr 2c l and x 2 R imply ðxyÞr 2c _ qd l, (F1e) ar 2c l implies ððx þ aÞy xyÞr 2c _ qd l.
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(3) (4)
if c = 0 and d = 1, then, l is the fuzzy ideal of R (see [1]); if c = 0 and d = 0.5, then l is the ð2; 2 _ qÞ-fuzzy ideal of R (see [3]); if c = 0.5 and d = 1, then l is the ð2; 2 _ qÞ-fuzzy ideal of R (see [11]); l is the fuzzy ideal of R with thresholds (c, d) (see [3]).
For any fuzzy set l of R, we define lcr ¼ fx 2 Rjxr 2c lg, ldr ¼ fx 2 Rjxr qd lg, and ½ldr ¼ fx 2 Rjxr 2c _ qd lg for all r 2 ½0; 1. It is clear that [l]dr = lcr [ ldr . The next theorem provides the relationship between ð2c ; 2c _ qd Þ-fuzzy ideals of R and crisp ideals of R. Theorem 3.2 (1) (2)
(3)
Let l be a fuzzy set of R. Then,
l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R if and only if lcr (=[) is an ideal of R for all r 2 ðc; d. If 2d = 1 ? c, then, l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R if and only if ldr (=[) is an ideal of R for all r 2 ðd; 1. If 2d = 1 ? c, then, l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R if and only if [l]dr (=[) is an ideal of R for all r 2 ðc; 1.
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Proof (1)
(2) (3)
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(2)
Let l be an ð2c ; 2c _ qd Þ-fuzzy ideal of R and x; y 2 lcr for all r 2 ðc; d, then, l(x) C r [ c and l(y) C r [ c. It follows that l(x ? y) _ c C l(x) ^ l (y) ^ d C r ^ d = r, that is, x þ y 2 lcr . Similarly, we can prove the other conditions of ideals hold. Hence, lcr is an ideal of R for all r 2 ðc; d. Conversely, assume that lcr is an ideal of R for all r 2 ðc; d. Let x; y 2 R. If l(x ? y) _ c \ r = l(x) ^ l (y) ^ d, then, xr 2c l; yr 2c l, but ðx þ yÞr 2c _ qd l, that is, x; y 2 lcr . Since lcr is an ideal, we have x þ y 2 lcr , a contradiction. Hence (F2a) holds. Similarly, we can prove (F2a’), (F2b), (F2c), (F2d), and (F2e) hold. Thus, l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R. The proof is similar to the proof of (1). Let l be an ð2c ; 2c _ qd Þ-fuzzy ideal of R and r 2 ðc; 1. Then, for all x; y 2 ½ldr , we have xr 2c _ qd l; yr 2c _ qd l, that is, l(x) C r [ c or l(x) [ 2d - r [ 2d - 1 = c, and l(y) C r [ c or l(y) [ 2d - r [ 2d - 1 = c. Since l is an ð2c ; 2c _ qd Þfuzzy ideal of R, then l(x ? y) _ c C l(x) ^ l (y) ^ d, and so l(x ? y) C l(x) ^ l (y) ^ d.
Case 1 r 2 ðc; d. Then, 2d - r C d C r and so l(x ? y) C r ^ r ^ d = r or l(x ? y) C r ^ (2d - r) ^ d = r or l(x ? y) C (2d - r) ^ (2d - r) ^ d = d C r. Hence ðx þ yÞr 2c l: Case 2 r 2 ðd; 1. Then, 2d - r \ d \ r and so l(x ? y) C r ^ r ^ d = d [ 2d - r or l(x ? y) [ r ^ (2d - r) ^ d = 2d - r or l(x ? y) [ (2d - r) ^ (2d - r) ^ d = 2d - r. Hence (x ? y)r qdl. Thus, in any case, ðx þ yÞr 2c _ qd l, i.e., x þ y 2 ½ldr . Similarly, we can prove the other conditions. Hence [l]dr is an ideal of R. Conversely, assume that [l]dr is an ideal of R for all r 2 ðc; d. Let x; y 2 R. If l(x ? y) _ c \ r = l(x) ^ l (y) ^ d, then xr 2c l; yr 2c l; but ðx þ yÞr 2c _ qd l;
(3)
Definition 3.2 A fuzzy set l of R is called an ð2c ; 2c _ qd Þ-fuzzy subnear-ring of R if for all t; r 2 ðc; 1 and x; y; a 2 R, (F3a) ðx þ yÞt ^ r 2c l implies xt 2c _ qd l or yr 2c _ qd l, (F3a’) ðxÞt 2c l implies xt 2c _ qd l, (F3b) ðxyÞt ^ r 2c implies xt 2c _ qd l or yr 2c _ qd l. Moreover, l is called an ð2c ; 2c _ qd Þ-fuzzy ideal of R if l is an ð2c ; 2c _ qd Þ-fuzzy subnear-ring of R and (F3c) ðy þ x yÞr 2c l implies xt 2c _ qd l, (F3d) ðxyÞr 2c l and x 2 R imply yr 2c _ qd l, (F3e) ððx þ aÞy xyÞr 2c l implies ar 2c _ qd l. Theorem 3.3 A fuzzy set l of R is an ð2c ; 2c _ qd Þ-fuzzy subnear-ring of R if and only if for any x; y; a 2 R, (F4a) l(x ? y) _ d C l(x) ^ l(y), (F4a’) l(- x) _ d C l(x), (F4b) l(xy) _ d C l(x) ^ l (y). Moreover, l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R if l satisfies the above conditions and (F4c) l(y ? x - y) _ d C l(x), (F4d) l(xy) _ d C l(y), (F4e) l((x ? a)y - xy) _ d C l(a). Proof
If we take c = 0 and d = 0.5 in Theorem 3.2, we can conclude the following result: Corollary 3.1 (1)
Let l be a fuzzy set of R. Then,
l is an ð2; 2 _ qÞ-fuzzy ideal of R if and only if lr(=[) is an ideal of R for all r 2 ð0; 0:5 (see [3]).
The proof is similar to the proof of Theorem 3.1. h
Remark 3.2 For any ð2c ; 2c _ qd Þ-fuzzy ideal l of R, we can conclude that if d = 0.5, then l is the ð2; 2 _ qÞfuzzy ideal of R (see [7]). Theorem 3.4 (1) (2)
that is, x; y 2 ½ldr . Since lcr is an ideal, we have x þ y 2 ½ldr , a contradiction. Hence (F2a) holds. Similarly, we can prove (F2a’), (F2b), (F2c), (F2d), and (F2e) hold. Thus, l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R. h
l is an ð2; 2 _ qÞ-fuzzy ideal of R if and only if Q(l;r)(=[) is an ideal of R for all r 2 ð0:5; 1, where Qðl; rÞ ¼ fx 2 Rjxr q lg. l is an ð2; 2 _ qÞ-fuzzy ideal of R if and only if [l]r(=[) is an ideal of R for all r 2 ð0; 1 (see [7]).
l if l if
Proof
Let l be a fuzzy set of R. Then
is an ð2c ; 2c _ qd Þ-fuzzy ideal of R if and only lcr (=[) is an ideal of R for all r 2 ðd; 1. is an ð2c ; 2c _ qd Þ-fuzzy ideal of R if and only ldr (=[) is an ideal of R for all r 2 ðc; d. The proof is similar to the proof of Theorem 3.2. h
If we take d = 0.5 in Theorem 3.4, we can conclude the following result: Corollary 3.2 (1)
Let l be a fuzzy set of R. Then
l is an ð2; 2 _ qÞ-fuzzy ideal of R if and only if lr(=[) is an ideal of R for all r 2 ð0:5; 1 (see [7]).
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l is an ð2; 2 _ qÞ-fuzzy ideal of R if and only if Q(l;r)(=[) is an ideal of R for all r 2 ð0; 0:5.
If we take c = 0 and d = 0.5 in Theorem 4.2, we can conclude the following result:
4 (Strong) prime ð2c ; 2c _ qd Þ-fuzzy ideals In this Section, we introduce the concepts of prime and strong prime ð2c ; 2c _ qd Þ-fuzzy ideals of near-rings. In particular, we discuss the relationship between strong prime ð2c ; 2c _ qd Þ-fuzzy ideals and prime ð2c ; 2c _ qd Þfuzzy ideals of near-rings. Definition 4.1 An ð2c ; 2c _ qd Þ-fuzzy ideal l of R is called prime if for all x; y 2 R and t 2 ðc; 1, we have (P) ðxyÞt 2c l ) xt 2c _ qd l or yt 2c _ qd l: Theorem 4.1 An ð2c ; 2c _ qd Þ-fuzzy ideal l of R is prime if for all x; y 2 R, it satisfies: (P’) l(x) _ l(y) _ c C l(xy) ^ d. Proof Let l be a prime ð2c ; 2c _ qd Þ-fuzzy ideal of R. If there exist x; y 2 R such that l(x) _ l(y) _ c \ t = l(xy) ^ d, then c\t d; ðxyÞt 2c l, but xt 2c l and yt 2c l. Since l(x) ? t \ 2t B 2d and l(y) ? t \ 2t B 2d, then xt qd l and yt qd l, and hence, we have xt 2c _ qd l and yt 2c _ qd l, which is a contradiction. Thus, (P’) holds. Conversely, suppose that condition (P’) holds. Let ðxyÞt 2c l. Then, l(xy) C t and so l(x) _ l(y) C l(xy) ^ d C t ^ d. We consider the following two cases: If t 2 ðc; d, then l(x) C t or l(y) C t, that is, xt 2c l or yt 2c l. Thus, xt 2c _ qd l or yt 2c _ qd l. (ii) If t 2 ðd; 1, then l(x) _ l(y) C d, and so, l(x) C d or l(y) C d. Hence, l(x) ? t [ 2d or l(y) ? t [ 2d, that is, xtqdl or ytqdl. Thus, xt 2c _ qd l or yt 2c _ qd l. (i)
This proves that l is prime.
h
Theorem 4.2 An ð2c ; 2c _ qd Þ-fuzzy ideal l of R is prime if and only if lct (=[) is a prime ideal of R for all t 2 ðc; d. Proof Let l be a prime ð2c ; 2c _ qd Þ-fuzzy ideal of R and t 2 ðc; d. Then, by Theorem 3.2, lct is an ideal of R for all c \ t B d. Let xy 2 lct . By Theorem 4.1, we have l(x) _ l(y) C l(xy) ^ d C t ^ d = t, and so l(x) C t or l(y) C t. Thus, x 2 lct or y 2 lct . This shows that lct is a prime ideal of R, for all t 2 ðc; d. Conversely, assume that lct (=[) is a prime ideal of R for all t 2 ðc; d. Then by Theorem 3.2, l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R. Let ðxyÞt 2c l. Then xy 2 lct . Since lct is prime, x 2 lct or y 2 lct , that is, xt 2
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l or yt 2 l. Thus, xt 2c _ qd l or yt 2c _ qd l. Therefore, l must be a prime ð2c ; 2c _ qd Þ-fuzzy ideal of R. h
Corollary 4.1 Let l be a fuzzy set of R. Then l is a prime ð2; 2 _ qÞ-fuzzy ideal of R if and only if lt(=[) is a prime ideal of R for all t 2 ð0; 0:5 (see [7]). Theorem 4.3 If 2d = 1 ? c, then a fuzzy set l of R is a prime ð2c ; 2c _ qd Þ-fuzzy ideal if and only if ldr (=[) is a prime ideal of R for all t 2 ðd; 1. Proof Let l be a prime ð2c ; 2c _ qd Þ-fuzzy ideal of R and t 2 ðd; 1. Then by Theorem 3.2(2), ldt is an ideal of R and t [ d [ 2d - t. To prove ldr is prime, let xy 2 ldt . Hence l(xy) [ 2d - t. Since l is a prime ð2c ; 2c _ qd Þfuzzy ideal of R, we have l(x) _ l(y) _ c C l(xy) ^ d [ 2d - t C 2d - 1 = c. It follows that l(x) _ l(y) C 2d - t, and so x 2 ldt or y 2 ldt : Therefore, ldr is a prime ideal of R. Conversely, let ldr (=[) be a prime ideal of R for all t 2 ðc; 1, then by Theorem 3.2(2), we know l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R. Now, if there exist x; y 2 R such that l(x) _ l(y) _ c \ t = l(xy) ^ d. Then, t B d, l(xy) C t, l(x) \ t, l(x) ? t \ 2t B 2d, l(y) \ t, and l(y) ? t \ 2t B 2d. It follows that ðxyÞt 2c l, but xt 2c _ qd l and yt 2c _ qd l, a contradiction. Therefore, l(x) _ l(y) _ c C l(xy) ^ d for all x; y 2 R and so l is a prime ð2c ; 2c _ qd Þ-fuzzy ideal of R. h Theorem 4.4 If 2d = 1 ? c, then a fuzzy set l of R is a prime ð2c ; 2c _ qd Þ-fuzzy ideal if and only if [l]dt (=[) is a prime ideal of R for all t 2 ðc; 1. Proof Let l be a prime ð2c ; 2c _ qd Þ-fuzzy ideal of R. Then by Theorem 3.2(3), [l]dt is an ideal of R for all t 2 ðc; 1. To prove [l]dt is prime, let xy 2 ½ldt . Since [l]dt = h. ldt [ lct , we have xy 2 ldt or xy 2 lct : Case 1 xy 2 ldt lct . Then, l(xy) ? t [ 2d and l(xy) \ t. (1)
(2)
If l(xy) B d, then l(x) _ l(y) ? t C l(xy) ^ d ? t = l(xy) ? t [ 2d, which implies, l(x) ? t [ 2d or l(y) ? t [ 2d, that is, x 2 ldt ½ldt or y 2 ldt ½ldt . If l(xy) [ d, then d \ l(xy) \ t. Thus, l(x) _ l(y) ? t C l(xy) ^ d ? t = d ? t [ 2d. Hence x 2 ldt ½ldt or y 2 ldt ½ldt :
Case 2 (1)
xy 2 lct . Then, l(xy) C t.
If t B d, then l(x) _ l(y) C l(xy) ^ d C t, which implies, x 2 lct ½ldt or y 2 lct ½ldt :
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(2)
If t [ d, then l(x) _ l(y) C t ^ d = d, which implies, l(x) _ l(y) ? t [ 2d. Hence, x 2 ldt ½ldt or y 2 ldt ½ldt :
Therefore, [l]dt is a prime ideal of R. Conversely, let [l]dt (=[) be a prime ideal of R for all t 2 ðc; 1, then by Theorem 3.2(3), we know l is an ð2c ; 2c _ qd Þ-fuzzy ideal of R. Let ðxyÞt 2c l, then xy 2 lct ½ldt . Since [l]dt is prime, we have x 2 ½ldt or y 2 ½ldt . This implies, xt 2c _ qd l or yt 2c _ qd l. Thereh fore, l is a prime ð2c ; 2c _ qd Þ-fuzzy ideal of R. If we take c = 0 and d = 0.5 in Theorem 4.4, we can conclude the following result: Corollary 4.2 Let l be a fuzzy set of R. Then, l is a prime ð2; 2 _ qÞ-fuzzy ideal of R if and only if [lt](=[) is a prime ideal of R for all t 2 ð0; 1 (see [7]). Now, we give the concept of strong prime ð2c ; 2c _ qd Þfuzzy ideals of near-rings. Definition 4.2 An ð2c ; 2c _ qd Þ-fuzzy ideal q of R is called strong prime if for every ð2c ; 2c _ qd Þ-fuzzy ideals l and m of R, it satisfies: (P’’) l m q implies l q or m q: Example 4.1 Let R = {0, a, b, c} be Klein’s four group. Define multiplication in N as follows:
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W l(x) C t. Hence, ðtI tJ ÞðxÞ ¼ x¼yz tI ðyÞ ^ tJ ðzÞ t lðxÞ. Therefore, tI tJ l. Since l is a strong prime ð2c ; 2c _ qd Þ-fuzzy ideal of R, we have tI l or tJ l, this implies I lct or J lct . This completes the proof. h The following is a consequence of Theorems 4.2 and 4.5. Theorem 4.6 Every strong prime ð2c ; 2c _ qd Þ-fuzzy ideal of a near-ring is a prime ð2c ; 2c _ qd Þ-fuzzy ideal. If we take c = 0 and d = 0.5 in Theorem 4.6, we can conclude the following result: Corollary 4.3 Every strong prime ð2; 2 _ qÞ-fuzzy ideal of a near-ring is a prime ð2; 2 _ qÞ-fuzzy ideal (see [10]). Remark 4.1 The converse of Theorem 4.6 is not true in general as shown in the following example. Example 4.2 Let ðZ; þ; Þ be the near-ring (it is also a ring) of all integers. Define fuzzy sets l, m and x of Z as follows: 8x 2 Z 8 x ¼ 0; < 1 lðxÞ ¼ 0:4 x 2 ð2Þnf0g; : otherwise 80 x ¼ 0; < 1 mðxÞ ¼ 0:4 x 2 ð3Þnf0g and : otherwise 80 x ¼ 0; < 1 xðxÞ ¼ 0:5 x 2 ð4Þnf0g; : 0 otherwise Then, l, m, and x are ð2c ; 2c _ qd Þ-fuzzy ideal of Z and non-empty subset lct is a prime ideal of R for all t 2 ðc; d. By Theorem 4.2, we know that l is a prime ð2c ; 2c _ qd Þfuzzy ideal of Z, but it is not strong prime. In fact, m x l, but m * l and x * l.
Then ðR; þ; Þ is a near-ring [8]. Define a fuzzy set l of R as follows: lð0Þ ¼ 0:7; lðaÞ ¼ 0:7; lðbÞ ¼ 0:4 and lðcÞ ¼ 0:3: Then l is a strong prime ð20:4 ; 20:4 _q0:7 Þ-fuzzy ideal of R. Theorem 4.5 Let l be a strong prime ð2c ; 2c _ qd Þ-fuzzy ideal of R. Then lct (=[) is a prime ideal of R for all t 2 ðc; d. Proof Let t 2 ðc; d be such that lct is non-empty. Then lct is an ideal of R by Theorem 3.2(1). Now we show that lct is prime. Let I and J be two ideals of R such that IJ lct . Then it is easy to see that tI and tJ are two ð2c ; 2c _ qd Þfuzzy ideals of R and that tI tJ l. In fact, let x 2 R. If (tI tJ)(x) = 0, then (tI tJ)(x) = 0 B l(x). Otherwise, there exist a; b 2 R such that x = ab and tI(a) ^ tJ(b) = 0. This implies a 2 I and b 2 J, hence x 2 IJ lct , that is,
Acknowledgments Supported by the National Natural Science Foundation of China (60875034); the Innovation Term of Education Committee of Hubei Province, China (T201109) and the Natural Science Foundation of Hubei Province, China (2009CDB340).
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