The lattice of fuzzy (left, right) ideals of a ring is modular

Fuzzy Sets and Systems 125 (2002) 209–214

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The lattice of fuzzy (left, right) ideals of a ring is modular  Qingde Zhang ∗ Department of Computer Science, Liaocheng Teachers University, Shandong 252059, People’s Republic of China Received 17 December 1998; received in revised form 3 December 1999; accepted 21 June 2000

Abstract We use the technique of nested set to study the structure of the lattice of all fuzzy (left, right) ideals of a given ring. We give a way to depict the supremum of fuzzy (left, right) ideals by nested set and prove that the lattice of all fuzzy (left, c 2002 Elsevier Science B.V. All rights reserved. right) ideals of a given ring is modular.
Keywords: Lattice; Fuzzy ideal; Supremum; Modularity

1. Introduction Since Rosenfeld [13] applied the notion of fuzzy sets to abstract algebra and introduced the notion of fuzzy subgroups, the literature of various fuzzy algebraic concepts has been growing very rapidly. However, the study of these algebraic structures from the point of lattice theory was initial. Recently, many authors have made some interesting studies of this theory and obtained a lot of good results [1–3,5,7,8,11,15]. Ajimal and Thomas [2] investigated the lattice structure of various sublattices of the lattice of fuzzy subrings of a given ring and proved that a special class of fuzzy ideals forms a modular sublattice of the lattice of fuzzy ideals of a ring. Zhang and Meng [15] carried on this work and proved that the lattice of all fuzzy (left, right) ideals with same tip “t” (i.e.,  Project proposed by the Foundation of Natural Science of Shandong Province. ∗ Corresponding author. E-mail address: [email protected] (Q. Zhang).

(0) = t) is modular. We ask: Is the lattice of all fuzzy (left, right) ideals of a given ring modular? This paper will answer the question. In this paper, we use the concept of nested set to investigate the lattice Fi (R) of fuzzy (left, right) ideals. In Section 3, we investigate the structure of supremum  ∨ , the minimal ideal which contains ideals  and , in the lattice Fi (R). In Section 4, we prove that the lattice Fi (R) and its quotient set lattice Fi (R)=∼ are all modular. Thus, we generalize the main results of [2,15]. 2. Preliminary For the purpose of reference we start with a presentation of a few basic deGnitions and results. For details we refer to [2 – 4,6,9,10,13]. Throughout this paper, we shall denote a ring with respect to two binary operations “+” and “·” by R (in general, we omit “·”). Denition 2.1 (Liu [9]). Let  be a fuzzy set in a ring R. Then  is called a fuzzy subring of R, if for all x; y

c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/02/$ - see front matter  PII: S 0 1 6 5 - 0 1 1 4 ( 0 0 ) 0 0 1 1 0 - X

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in R (1) (x − y)¿(x) ∧ (y), (2) (xy)¿(x) ∧ (y): Denition 2.2 (Liu [9]). Let  be a fuzzy subring of a ring R. Then  is called a fuzzy (left, right) ideal of R if ((xy)¿(y); (xy)¿(x)) (xy)¿(x) ∨ (y); for all x; y in R: From now on, we shall only discuss the theory for the fuzzy ideal, all results in this paper are right too for left and right ideals. Denition 2.3 (Ajimal and Thomas [2]). Let  and  be fuzzy sets in a ring R. Then their sum  +  is deGned as follows:
( + )(z) = {(x) ∧ (y) | x + y = z} for all z in R: Denition 2.4 (Liu [9]). Let  be a fuzzy set in a set S. Then, for  ∈ [0; 1], the sets  = {x | x ∈ S; (x)¿};  = {x | x ∈ S; (x)¿} · are called level cut and strong cut of , respectively.

Proposition 2.2. Let  be a fuzzy subset of R; then the following statements are equivalent: (1)  is a fuzzy subring (ideal) of R; (2)  are crisp subrings (ideals) of R for all 06 6(0); (3)  are crisp subrings (ideals) of R for all 06 · ¡(0). Proof. We prove (1) ⇔ (3) for fuzzy subrings only. Let  be a fuzzy subring of R. For any x; y ∈  · (06¡(0)), we have (x)¿ and (y)¿ by DeGnition 2.4, so (x − y)¿(x) ∧ (y)¿ and (xy)¿, and then x − y ∈  , xy ∈  . This means · · that  is a crisp subring of R. · Conversely, let  be crisp subrings of R for all · 06¡(0). For any x; y ∈ R, let (x) = s, (y) = t and s6t, then for any 0¡¡s, (x)¿s − , (y) ¿s − . This means that x; y ∈ − , from − are · · subrings we have x − y, xy ∈ − , and then (x − y) · ¿s − , (xy)¿s − . By the arbitrariness of  we have (x − y)¿s, (xy)¿s. Therefore,  is a fuzzy subring of R. Proposition 2.3 (Ajimal and Thomas [2]). F(R) forms a complete lattice under the ordering of fuzzy set inclusion 6. Proposition 2.4 (Ajimal and Thomas [2]). Fi (R) is a complete sublattice of F(R).

Proposition 2.1 (Luo [10]). Let  be a fuzzy subset of S; then  (1)  = 0661  (i:e:; (x) = { | x ∈  } for all x  in S).  (2)  = 0661  (i:e:; (x) = { | x ∈  } for · · all x in S).

Proposition 2.5 (Ajimal and Thomas [2]). Let Fi st (R) be the set of all fuzzy ideals with supproperty and same tip “t”; of a ring R; then Fi st (R) is modular sublattice of Fi (R).

For convenience, we introduce the following symbols:

Proposition 2.6. Let Fi t (R) be the set of all fuzzy ideals with same tip “t”; of a ring R; then Fi t (R) is modular sublattice of Fi (R).

F(R) = { |  is a fuzzy subring of R}; Fi (R) = { |  is a fuzzy ideal of R}: Obviously, for any  ∈ F(R); we have (x)6(0)

and

(x) = (− x)

for all x in R; we call (0) the tip of the fuzzy subring .

The author proved the following results in [15]:

Denition 2.5 (Luo [10]). Let S be a set and P(S) denote the power set of S. A map H : [0; 1] → P(S);  → H () is called a nested set of S, if 1 ¡2 ⇒ H (1 ) ⊇ H (2 ); denoted by H = {H () |  ∈ [0; 1]}.

Q. Zhang / Fuzzy Sets and Systems 125 (2002) 209–214

If H is a nested set of S, let  = H (); ∈[0;1]

 (i.e., (x) = { | x ∈ H ()} for all x in S); then  is a fuzzy subset of S, called the fuzzy subset determined by nested set H . Proposition 2.7 (Luo [10]). Let H be a nested set of set S;  is the fuzzy subset determined by H; then (1)  ⊆ H () ⊆  , ·   (2)  = ¡ H () (¿0);  = ¿H () (¡1). · Denition 2.6. Let ,  be two (left, right) ideals of R, we call the minimal ideal, which contains  and , the supremum of  and , denoted by  ∨ . 3. The structure of supremum In this section, we will discuss the relation between fuzzy ideal and the nested set, and investigate the structure of supremum  ∨  in the lattice Fi (R). Proposition 3.1. Let H be a nested set of R;  be the fuzzy subset of R determined by H . If H () are crisp ideals of R for all H () = ∅; then  is a fuzzy ideal of R. Proof. For any x; y in R, suppose (x) ∧ (y) = , then
(x) = { | x ∈ H ()}¿;
(y) = { | y ∈ H ()}¿: If ¿0, for any  ∈ (0; ), by the deGnition of nested set we have x ∈ H (−), y ∈ H ( − ), because H ( − ) is the crisp ideal, we have x − y ∈ H ( − ), thus (x − y)¿ − , by the arbitrariness of , we obtain (x − y)¿ = (x) ∧ (y). Similarly, (rx)¿(x), (xr)¿(x). If  = 0, it is obvious that the inequalities above are right. Therefore,  is a fuzzy ideal of R. From the proof above we can obtain the following result:

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Proposition 3.2. Let H be a nested set of R and  be the fuzzy subset of R determined by H . If H () are crisp subrings of R for all H () = ∅; then  is a fuzzy subring of R. Remark 3.3. The converse of Proposition 3.1 (3:2) is not true. For example: Let Z be the integer ring, S be the even number ring. Now, we deGne the nested set H as follows:  06¡0:5;   S; H () = {2; 4};  = 0:5;   ∅; ¿0:5: It is easy to verify that H is a nested set of Z and the fuzzy subset  determined by H is (x) =




{ | x ∈ H ()} =

0:5;

x ∈ S;

0;

x ∈= S:

Obviously,  is a fuzzy ideal of R, but H (0:5) = {2; 4} is not a crisp ideal of R. The author discussed the structure of supremum  ∨  in the lattice Fi t (R) and obtained Proposition 3.4 (Zhang and Meng [15]). In the lattice Fi t (R);  ∨  =  + . Obviously; ( ∨ )(0)¿(0) ∨ (0); ( + )(0)6 (0) ∧ (0); so Proposition 3:4 is not true in the lattice Fi (R). The following theorem gives the structure of  ∨  in the lattice Fi (R) (of course; it is suitable to Fi t (R)). Theorem 3.5. Let ;  ∈ Fi (R); (0) = s; (0) = t. If t¿s; then  ∨  is determined by the following nested set:   +  ; ¡s;   ·  · s6¡t; H () =  ; ·    ∅; t6: Proof. Let  be the fuzzy subset of R determined by the nested set H . From ;  ∈ Fi (R) we know that  + ·  and  are all crisp ideals (or ∅) of R. By Proposition · · 3.1  is a fuzzy ideal of R.

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If ¡s,  ⊆  + = H (); If s6,  = ∅ ⊆H (), · · · · hence   =  ⊆ H () = ; · 0661

0661

(by Propositions 2:1 and 2:7): Similarly, we can obtain  ⊆ , therefore  ∨  ⊆ . If $ is any fuzzy ideal of R such that  ⊆ $ and  ⊆ $, then  ⊆ $ and  ⊆ $ for all  ∈ [0; 1], thus · · · · H () ⊆ $ for all  ∈ [0; 1], and then it follows that ·   = H () ⊆ $ = $: · 0661

0661

This implies that  is the minimal fuzzy ideal which contains  and , i.e.,  =  ∨ . This completes the proof. Corollary 3.6. Let ;  ∈ Fi (R); (0) = s; (0) = t. If t¿s; then ( ∨ )(0) = t and   · + · ; ¡s; s6¡t; ( ∨ ) =  ; ·  · ∅; t6: Proof. (1) Let H be a nested set same as in Theorem 3.5, then

( ∨ )(0) = { | 0 ∈ H ()} = { | ¡t} = t: (2) Since , 6 ∨ , so  ,  ⊆ ( ∨ ) . · · · If ¡s,  + =  ∨  ⊆ ( ∨ ) ; also ( ∨ ) ⊆ · · · · · · H () =  +  . Therefore, ( ∨ ) =  +  . · · · · · If s6¡t, ( ∨) ⊆ H ()=  , hence ( ∨) = . · · · · If t6, ( ∨ ) = ∅. · This completes the proof. 4. Modularity Denition 4.1. A lattice L is said to be modular, if it satisGes the following condition: a¿b ⇒ a ∧ (b ∨ c) = b ∨ (a ∧ c): In this section, we shall prove that Fi (R) is modular. Then we introduce an equivalence relation ∼ on Fi (R) and prove that the quotient set Fi (R)=∼ forms a modular lattice.

Lemma 4.1. The lattice of all crisp ideals of a ring R is modular and I ∨ J = I + J;

I ∧ J = I ∩ J;

where I; J are the crisp ideals of R. Lemma 4.2. Let ;  ∈ Fi (R); then  ∧  is determined by the following nested set: H () = ( ∧ ) =  ∧  =  ∩  : · · · · · Theorem 4.3. Fi (R) is modular. Proof. For the triplet ; ; " ∈ Fi (R) with ¿, let (0) = a, (0) = b, "(0) = c, then a¿b. Next, we prove  ∧ ( ∨ ") =  ∨ ( ∧ "): (1) If c¿a¿b, then  ∨ " is determined by the following nested set:   + " ; ¡b;   ·   " ·; b6¡a;  · H () =  a6¡c; " ;    · ∅; c6 and then,  ∧ ( ∨ ") is determined by the following nested set:   ∧ ( + " ); ¡b;   · ·  · b6¡a;  ∧ " ; · · K() =  a6¡c;   ∅; ∅; c6:  ∧ " is determined by the following nested set:   ∧ " ; ¡b;  ·   ·  ∧ " ; b6¡a;  · · H () =  a6¡c;   ∅; ∅; c6;  ∨ ( ∧ ") is determined by the following nested set (note: ( ∧ ")(0) = a):   + ( ∧ " ); ¡b;  · ·   ·  ∧ " ; b6¡a; · · K  () =  ∅; a6¡c;   ∅; c6:

Q. Zhang / Fuzzy Sets and Systems 125 (2002) 209–214

From Lemma 4.1 and  ⊇  , we can obtain · · K() = K  () for all  ∈ [0; 1]; thus, we have  ∧ ( ∨ ") =  ∨ ( ∧ "): (2) If a¿b¿c, then  ∨ " is determined by the following nested set:   + " ; ¡c;   ·  · c6¡b;  ; · H () =  ∅; b6¡a;   ∅; a6; and then,  ∧ ( ∨ ") is determined by the following nested set:   ∧ ( + " ); ¡c;   · ·  · c6¡b;  ∧  ; · · K() =  ∅; b6¡a;   ∅; a6:  ∧ " is determined by the following nested set:   ∧ " ;   ·  · ∅;  H () =  ∅;   ∅;

¡c; c6¡b; b6¡a; a6;

 ∨ ( ∧ ") is determined by the following nested set:   + ( ∧ " ); ¡c;  · ·   · c6¡b;  ;  · K () =  b6¡a;   ∅; ∅; a6: It is easy to know that K() = K  () for all  ∈ [0; 1] (note  ⊇  ); therefore, we have · ·  ∧ ( ∨ ") =  ∨ ( ∧ "): (3) If a¿c¿b, then  ∨ " is determined by the following nested set:   + " ; ¡b;   · ·  b6¡c; " ; · H () =  ∅; c6¡a;   ∅; a6;

213

and then,  ∧ ( ∨ ") is determined by the following nested set:   ∧ ( + " ); ¡b;   · · ·    ∧" ; b6¡c;   · · K() =  c6¡a;  ∅;    ∅; a6:  ∧ " is determined by the following nested set:   ∧ " ; ¡b;   · ·     ∧ " ; b6¡c;    · · H () =  ∅; c6¡a;     ∅; a6 and then,  ∨ ( ∧ ") is determined by the following nested set:   + ( ∧ " ); ¡b;   · · ·    ∧" ; b6¡c;   · · K  () =  c6¡a;  ∅;    ∅; a6: Obviously, K() = K  () for all  ∈ [0; 1]; hence,  ∧ ( ∨ ") =  ∨ ( ∧ "): To sum up (1) – (3) for any case we have  ∧ ( ∨ ") =  ∨ ( ∧ "): This completes the proof. Corollary 4.4 (Ajimal and Thomas [2]; Zhang and Meng [15]). Fist (R) and Fit (R) are all modular. Lastly; we discuss the relation between Fi (R) and Fit (R). We deGne the binary relation ∼ on Fi (R) as follows:  ∼  ⇔ (0) = (0): It is easy to verify that ∼ is an equivalence relation. For any  ∈ Fi (R); let [] = { |  ∈ Fi (R) and (0) = (0)} = Fi(0) (R); then the quotient set determined by ∼ is Fi (R)= ∼ = {[] |  ∈ Fi (R)} = {Fit (R) | t ∈ [0; 1]}:

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Theorem 4.5. De5ne the operation ∨; ∧ as follows:

References

[] ∨ [] = [ ∨ ];

[1] N. Ajimal, K.V. Thomas, The lattice of fuzzy subgroups and fuzzy normal subgroups, Inform. Sci. 76 (1994) 1–11. [2] N. Ajimal, K.V. Thomas, The lattices of fuzzy ideals of a ring R, Fuzzy Sets and Systems 74 (1995) 371–379. [3] N. Ajimal, K.V. Thomas, The lattice of fuzzy normal subgroups is modular, Inform. Sci. 83 (1995) 199–209. [4] V.N. Dixit, Level subgroups and union of fuzzy subgroups, Fuzzy Sets and Systems 37 (1990) 359–371. [5] K.C. Gupta, S. Ray, Modularity of the quasihamiltonian fuzzy subgroups, Inform. Sci. 79 (1994) 233–250. [6] N. Jacobson, Basic Algebra I, Freeman, San Francisco, CA, 1974. [7] J.-G. Kim, Fuzzy orders relative to fuzzy subgroups, Inform. Sci. 80 (1994) 341–348. [8] J.-G. Kim, S.-J. Cho, Structure of a lattice of fuzzy subgroups, Fuzzy Sets and Systems 89 (1997) 263–266. [9] W.-J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133–139. [10] Luo Chengzhong, A introduction to fuzzy sets, Beijing Teachers University Publishing House, Beijing, 1989. [11] V. Murali, Lattice of fuzzy subalgebras and closure system in IX , Fuzzy Sets and Systems 41 (1991) 101–111. [12] S. Ray, The lattice of all idempotent fuzzy subsets of a groupoid, Fuzzy Sets and Systems 96 (1998) 239–245. [13] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512–517. [14] U.M. Sway, D.V. Raju, Fuzzy ideals and congruences of lattices, Fuzzy Sets and Systems 95 (1998) 249–253. [15] Qingde Zhang, Guangwu Meng, On the lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 112 (2000) 349–353.

[] ∧ [] = [ ∧ ]:

Then (Fi (R)=∼; ∨; ∧) forms a modular lattice. Proof. Obviously, the deGnitions of sup ∨ and inf ∧ are reasonable. By the natural map from Fi (R)=∼ to Fi (R) and Theorem 4.3, we can easily obtain that (Fi (R)=∼; ∨; ∧) forms a modular lattice. It is well known that the lattice of all ideals of a ring is modular, which is important in the classical ring theory. Many papers have studied the lattice of all fuzzy ideals of a ring Fi (R) and proved that some of its special sublattices are modular, but whether the lattice Fi (R) is modular is still been unsolved. Obviously, the study of modularity of Fi (R) is of great value to the fuzzy ring theory. The result achieved in this paper will lay a solid foundation in studying the structure and mutual relations of fuzzy ideals. Moreover, the method applied in this paper has oNered a new approach to the study of fuzzy algebra. Acknowledgements The author is very grateful to the referees for their valuable comments on the Grst version of this paper.