n−Primary Submodules IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 Fuzzy Semimaximal ideals I, M.A.Hadi , M. A.Hamil Department of Mathematics, College of Education Ibn-Al-Haitham University of Baghdad Abstract Let R be a commutative ring with identity. A proper ideal I of R is called semimaximal if I is a finite intersection of maximal ideals of R. In this paper we fuzzify this concept to fuzzy ideals of R, where a fuzzy ideal A of R is called semimaximal if A is a finite intersection of fuzzy maximal ideals. Various basic properties are given. Moreover some examples are given to illustrate this concept. Introduction Let R be a commutative ring with unity. It is well-known that a proper ideal M of a ring R is called maximal if for every ideal B of R, M  B  R implies B=R. D.S.Malik and J.N.Mordeson in (1) introduced and studied the concept of fuzzy maximal ideal of R, where a fuzzy ideal A of R is maximal if 1. A is not constant, 2. for any fuzzy ideal B of R if A  B, then either A = B or B = R In fact D.S.Malik and J.N.Mordeson in (1) explained that a fuzzy maximal ideal A on R can not be defined as a fuzzy ideal A  R such that for each fuzzy ideal B of R, if A  B  R implies B = R. Goodreal in (2) introduced the concept of semimaximal ideals, where an ideal I of R is called semimaximal if it is a finite intersection of maximal ideals. Also, this concept was studied by Hatem in (3). In this paper, we fuzzify this concept to fuzzy ideals of R, where a fuzzy ideal A of R is called semimaximal if A is a finite intersection of fuzzy maximal ideals of R. Moreover, we generalize many properties of maximal and semimaximal ideals in to fuzzy semimaximal ideals of a ring. This paper consists of four sections. In S.1, we recall many definitions and properties which are needed in our work. In S.2, Various basic properties about fuzzy semimaximal ideals are discussed. In S.3, the image and inverse image of fuzzy semimaximal ideals are studied. In S.4, we study the behavior of fuzzy semimaximal ideals in a ring R, where R = R1R2 (direct sum of two rings R1, R2). S.1 Preliminaries This section contains some definitions and properties of fuzzy subset, fuzzy ideals and fuzzy rings, which we used in the next section. First we give some basic definitions and properties of fuzzy subsets. Let R be a commutative ring with unity, A fuzzy subset of R is a function from R into [0,1], (4). A fuzzy subset A is called a fuzzy constant if A(x) = t,  x R, t  [0,1], (4). For each t  [0,1], the set At = {x R, A(x)  t} is called a level subset of A, (4). A denoted the set {x  R,A(x) = A(0)}, (1). If x R and t  [0,1], we let xt denote the fuzzy subset of A define by xt(y) = 0 if x  y and xt(y) = t if x = y, xt is called a fuzzy singleton, (5). If A and B are fuzzy subsets of R, then: IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 1. A  B if A(x)  B(x), for all x R, (6), 2. A = B if A(x) = B(x), for all x R, (6). We define A  B by, (A  B)(x) = min{A(x), B(x)},  x R, and let {A:} be a collection of fuzzy subsets of R. Define the fuzzy subset of R (intersection) by ( )    x = inf {A:}, for all x R, (6). Let A and B be fuzzy subsets of R, then for all t [0,1], (AB)t=At Bt,(3). Let f be a mapping from a set M into a set N. Let A be a fuzzy subset of M and B be a fuzzy subset of N. The image of A denoted by f (A) is the fuzzy subset of N defined by: 1 1 sup { ( ) ( ) ,if (y) , for all }, ( ) 0 otherwise               z z f y f y f where f -1 (y) ={x  M, f (x) = y}. And the inverse image of B, denoted by f -1 (B) is the fuzzy subset of M, denoted by f -1 (B)(x) = B(f (x)), for all x  M, (4). Let f be a function from a set M into a set N. A fuzzy subset A of M is called f – invariant if A(x) = A(y), whenever f (x) = f (y) where x, y  M, (7). If f a function from a set M into a set N, A1 and A2 are fuzzy subsets of M and B1, B2 are fuzzy subsets of N, then 1. f (A1A2) = f (A1)  f (A2), whenever A1, A2 f –invariant, (8) 2. f -1 (B1B2) = f -1 (B1)  f -1 (B2), (8). Moreover the following definitions and properties are needed later Definition 1.1: (9) A fuzzy subset K of R is called a fuzzy ideal of R if for each x, y  R, then: 1. K(x – y)  min {K(x), K(y)}, 2. K(x y)  max {K(x), K(y)}. Definition 1.2: (10) Let X be a fuzzy subset of a ring R, then X is called fuzzy ring of R if for each x, y  R, then 1. X  0, 2. X(x – y)  min {X(x), X(y)}, 3. X(x y)  max {X(x), X(y)}. Proposition 1.3: (1) Let {A,} be a family of fuzzy ideals of R, then     is a fuzzy ideal of R. Definition 1.4: (11) Let X be a fuzzy ring of a ring R, let A be a fuzzy subset of X such that A  X. Then A is called a fuzzy ideal of a fuzzy ring X if for each x, y  R 1. A(x – y)  min {A(x), A(y)}, 2. A(x y)  min{max {A(x), A(y), X(x y)}. Note 1.5: It is clear that any fuzzy ideal of a ring R is a fuzzy ideal of a fuzzy ring X of R such that X(a) = 1,  a  R. Proposition 1.6: Let f be a homomorphism from a ring R1 into a ring R2, then the following are true: 1. f (A) is a fuzzy ideal of R2, for each fuzzy ideal A of R1, (6). 2. f -1 (B) is a fuzzy ideal of R1, for each fuzzy ideal B of R2, (7) Proposition 1.7: (1) IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 Let A, B be fuzzy ideals of a ring R such that A(0) = 1 = B(0). Then (AB) = A  B. Proposition 1.8: (1) Let {A:} be a family of fuzzy ideals of R such that A(0) = 1, for all . Then ( )      =        . S.2 Basic Properties of Fuzzy Semimaximal Ideals First, we give the following lemma which summarized the basic properties of fuzzy maximal ideals. Lemma 2.1: (1) 1. Let A be a fuzzy maximal ideal of R, then A(0) = 1 (see Th.3.3). 2. Let A be a fuzzy maximal ideal of R, then Im(A) = 2; That is A is a two valued, where Im(A) denotes image of A and Im(A) denotes the cardinality of Im(A) (see Th.3.4). 3. If A is a fuzzy maximal ideal of R, then A is a maximal ideal of R (see Th.3.5). 4. If A is a fuzzy ideal of R and A is a maximal ideal of R, then A is two valued (see Th. 3.6). 5. If A is a fuzzy ideal of R and A is a maximal ideal of R such that A(0) = 1. Then A is a fuzzy maximal ideal of R (see Th. 3.7). 6. If I  R be an ideal of R. Then I is a maximal ideal of R if and only if I is a fuzzy maximal ideal of R (see Cor. 3.8). Thus we introduce the following: Definition 2.2: Let A be a fuzzy ideal of R, A is called a fuzzy semimaximal ideal if A is a finite intersection of fuzzy maximal ideals of R. Remarks 2.3: 1. It is clear that every fuzzy maximal ideal is fuzzy semimaximal ideal. However the converse is not true as the following example shows: Let A : Z  [0,1] defined by: 1 6 , ( ) 1 otherwise 2         x x It is clear that A is fuzzy ideal of Z and A is not fuzzy maximal ideal since A = 6Z is not maximal ideal (see Lemma 2.1(3)). However A = A1  A2, where A1 and A2 are fuzzy ideals defined by: A1:Z  [0,1] A2:Z  [0,1] 1 1 2 , ( ) 0 otherwise       x x 2 1 3 , ( ) 0 otherwise       x x A1 and A2 are fuzzy maximal ideals of Z since (A1) = 2Z and (A1) = 3Z are maximal ideals (see Lemma 2.1(5)). 2. If A is a fuzzy semimaximal ideal of R, then A(0) = 1. Proof. Since A is a fuzzy semimaximal ideal of R, A = A1  A2   An, where Ai is a fuzzy maximal ideal of R, for all i = 1, 2, , n. Since Ai (0) = 1 by lemma 2.1(1), then A(0) = 1 (0)   n i i = min{ Ai (0), i = 1, 2, , n} = 1 3. If A and B are fuzzy semimaximal ideals of R, then A  B is a fuzzy semimaximal ideal of R. IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 Proof. Since A and B are fuzzy semimaximal ideal of R, then A = 1  n i i , B = 1  m i i , where Ai is a fuzzy maximal ideal, for all i = 1, 2, , n and Bi is a fuzzy maximal ideal, for all i = 1, 2, , m. Thus A  B = A1  A2   An  B1  B2   Bm; That is A  B is a finite intersection of fuzzy maximal ideals of R. 4. If { Ai, i = 1, 2, , n} be a family of fuzzy semimaximal ideals of R, then 1  n i i is a fuzzy semimaximal ideal of R. Proof. It is easy, so it is omitted. Compare the following result with lemma 2.1(3) Proposition 2.4: If A is a fuzzy semimaximal ideal of R, then A is semimaximal ideal of R. Proof. Since A is a fuzzy semimaximal ideal, so A = 1  n i i , where Ai is a fuzzy maximal ideal for all i = 1, 2, , n. Since Ai (0) = 1 (by Lemma2.1(1)), so that A = 1        n i i =   1    n i i (by prop.1.8) But (Ai) is maximal ideal,  i = 1, 2, , n by lemma 2.1(3). Hence A =   1    n i i . Thus A is a maximal ideal. The converse of this proposition is not true in general. However an example which will explain this depend on theorem 2.10. So we shall give it later (see Remark 2.11). Before giving our next result, we need to recall the following: Definition 2.5: (1) Let A be a fuzzy ideal of R, then A is called fuzzy prime if either A = R or 1. A is not constant and 2. For any fuzzy ideals B and C of R, if BC  A, then either B A or C  A. Definition 2.6: (1) Let A be a fuzzy ideal of R. The fuzzy radical of A denoted by  defined by  =  { P:P  £(A)}, where £(A) denotes the set of all fuzzy prime ideals of R which contains A. Proposition 2.7: If A is a fuzzy semimaximal ideal of R, then  = A Proof. Since A is a fuzzy semimaximal ideal, then A = A1  A2   An, where A1, A2, , An are fuzzy maximal ideals of R. But for each i = 1, 2, , n Ai is a fuzzy prime ideal, hence  i =Ai by (Theorem 5.13,(1)). Thus  = 1  n i i and so  = A Remark 2.8: If A is a fuzzy semimaximal ideal, then it is not necessary that A is a fuzzy prime ideal. We can give the following example: Example: Let A :Z  [0,1] defined by IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 1 0, ( ) 0 otherwise      x x By (Theorem 2.4(12)). A is a fuzzy prime ideal of Z, but A = (0) is not a semimaximal ideal in Z. Thus A is not fuzzy semimaximal ideal (by prop.2.4). Compare the following with (Lemma 2.1(6)). Proposition 2.9: Let I be an ideal of R, then I is a semimaximal ideal of R if and only if I is a fuzzy semimaximal ideal of R, where 1 , ( ) 0 otherwise      x x Proof. Since I is a semimaximal ideal, then 1    n i i , Ii is a maximal ideal,  i = 1, 2, , n. It is clear that 1 2       n     . But for each i = 1, 2, , n,   i  = Ii so 1 2 , ,   n    are fuzzy maximal ideals by (Lemma 2.1(6)).Thus 1     n i i   is a fuzzy semimaximal ideal. Conversely; If I is a fuzzy semimaximal ideal of R, then by (Lemma 2.1(3)), (I) is semimaximal, and since (I) = I. So the result is obtained. Compare the following with (Lemma.2.1(2)). Theorem 2.10: Let A be a fuzzy semimaximal ideal, then Im A = 2. Proof. 1  Im A since A(0) = 1 (by Rem.2.3(2)). We claim that for any 0  t < 1, At = R. Since A is a fuzzy semimaximal ideal, then A = A1  A2   An, where A1, A2, , An are fuzzy maximal ideals of R. Since 0  t < 1, then by the same proof of theorem 3.4 (1), we have (A1)t = (A2)t =  = (An)t = R But At = (A1)t  (A2)t    (An)t , so At = R, for all t, 0  t < 1. Suppose there exist t1, t2  [0,1], t1, t2  Im A. Then 1 2    t t = R which implies t1 = t2. Thus Im A has two valued namely 1, t. Remark 2.11: By using theorem 2.10, we can give an example which explains that the converse of proposition 2.4 is not true in general. Example: Let A :Z  [0,1] defined by 1 6 , 1 ( ) 2 6 , 2 0 otherwise             x x x A is not a fuzzy semimaximal ideal, since Im A = 3. However A(0) = 1, A = 6Z is a semimaximal ideal of Z. IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 Remark 2.12: If A a fuzzy semimaximal ideal of R and B is a fuzzy ideal of R such that B  R and A  B. Then it is not necessary that B is a fuzzy semimaximal ideal. Example: Let A :Z  [0,1] defined by 1 6 , ( ) 1 otherwise 2         x x A is a fuzzy semimaximal ideal (see Remark 2.3(1)). Let B :Z  [0,1] defined by 1 6 , 3 ( ) 2 6 , 4 1 otherwise 2               x x x It is clear that A  B. However Im A = 3, which implies that B is not a fuzzy semimaximal ideal, by theorem 2.10. Remark 2.13: If A a fuzzy semimaximal ideal and t  [0,1), then At does’nt need to be a semimaximal ideal of R. As can be seen by the following example: Example: Let A :Z  [0,1] defined by 1 2 , ( ) 1 otherwise 2         x x A is a fuzzy semimaximal ideal of Z and A = 2Z which is amaximal ideal and A(0) = 1, this implies that A is a fuzzy maximal ideal, so it is semimaximal. But A1/2 = {x:A(x)  1 2 } = Z which is not a semimaximal ideal. Recall that, the fuzzy Jacobson radical of a ring R denoted by F-J(R) is the intersection of all fuzzy maximal ideals of R (1). F-J(R) does’nt need to be a fuzzy semimaximal ideal of R. Example: Let {Ai , i = 1, 2, , n} be the collection of all fuzzy maximal ideals of Z, where 1 , ( ) 1 . i          i x p x x p , p is a prime number. IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 s a prime no. 1 s a prime no. 1 , F - J(R) 1 inf{ , } . i               p i i i p i x p i x p s a prime no. 1 s a prime no. 1 , F - J(R) 0 .             p i i i p i x p x p = 01 which is not a fuzzy semimaximal ideal of Z. Now, let F-J(R) denotes the intersection of all fuzzy semimaximal ideals of R. Then F- J(R) is called a fuzzy semijacobson radical of R. Remark 2.14: F-J(R) = F-J(R). Proof. It is clear that F-J(R)  F-J(R). Let xt  F-J(R). Then xt belongs to any fuzzy maximal ideal. Since any fuzzy semimaximal ideal A of R is a finite intersection of fuzzy maximal ideals, so xt  A. It follows that xt  F-J(R). Thus F-J(R) = F-J(R). S.3 Image and Inverse Image of Fuzzy Semimaximal Ideals In this section, we consider the homomorphic image and inverse image of fuzzy semimaximal ideals. Theorem 3.1: Let R1, R2 be two rings, let f : R1  R2 be an epimorphisim and every fuzzy ideal of R1 is f-invariant. Then if A is a fuzzy semimaximal ideal of R1, then f (A) is a fuzzy semimaximal of R2. Proof. A is a fuzzy semimaximal ideal of R1, then A = A1  A2   An, where A1, A2, , An are fuzzy maximal ideals of R1. Also, since every fuzzy ideal of R1 is f-invariant. So f (A) = f (A1  A2  An) = f (A1)  f (A2)   f (An). On the other hand, f (Ai) is a fuzzy maximal ideal of R2,  i = 1, 2, , n by (Th. 3.2 (1)) in(13) and note 1.5. Hence f (A) is a finite intersection of fuzzy maximal ideals. Thus f (A) is a fuzzy semimaximal ideal of R2. Theorem 3.2: Let R1, R2 be two rings, let f : R1  R2 be an epimorphisim. If B is a fuzzy semimaximal ideal of R2, then f -1 (B) is a fuzzy semimaximal ideal of R1. Proof. Since B is a fuzzy semimaximal ideal of R2, B = B1  B2   Bn, where Bi is a fuzzy maximal ideals of R2, for all i = 1, 2, , n. But f -1 (B) = f -1 (B1  B2  Bn) = f -1 (B1)  f -1 (B2)   f -1 (Bn) But for each i = 1, 2, , n, f -1 (Bi) is a fuzzy maximal ideal of R1 by (Th. 3.2)(2) in (13) and note 1.5. Hence f -1 (B) is a finite intersection of fuzzy maximal ideals. Thus f -1 (B) is a fuzzy semimaximal ideal of R1. S.4 Direct Sum of Fuzzy Semimaximal Ideals IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 In this section, we turn out attention to study fuzzy semimaximal ideals and direct sum. First we give the following lemmas which are useful in our work. Lemma 4.1: Let R1, R2 be two rings, let A, B be fuzzy ideals of R1, R2 respectively. Then AB is a fuzzy ideal of R1R2, where (A  B)(a,b) = min{A(a),B(b)}, for all (a,b)  R1  R2. Proof. By using note 1.5 and (Th.2.4.1.8)(14) the result follows directly. Lemma 4.2: Let R1, R2 be two rings, let A be a fuzzy ideals of R1  R2, then there exist fuzzy ideals B1 and B2 of R1, R2 respectively such that A = B1  B2. Proof. By using note 1.5 and (Th.2.4.1.9)(14) the result is obtained. Lemma 4.3: If A and B are fuzzy ideals of rings R1, R2 respectively then (A  B) = A  B. Proof. Let (x,y)  (A  B), then (A  B)(x,y) = 1 and so min{A(x),B(y)} = 1. This implies that A(x) = 1, B(y) = 1. Hence x  A and y  B. Thus (x,y)  A  B, so (A  B)  A  B. Conversely; Let (x,y) A  B. Then x  A and y  B. Hence A(x) = 1, B(y)=1. Thus min{A(x),B(y)} = 1 and so (A B)(x,y) = 1; That is (x,y)(AB). Thus (A  B)  A  B and hence (A  B)=A  B. It is known that (see (15)p.53): If R1, R2 be rings, R = R1  R2 and A is an ideal of R then A is a maximal ideal of R iff A = A1  R2 or A = R1  A2, where A1 is a maximal ideal of R1, A2 is a maximal ideal of R2. We generalize this result, to the following: Lemma 4.4: Let R1, R2 be two rings, R = R1  R2 and A is a fuzzy ideal of R then A is a fuzzy maximal ideal of R if and only if either A = B  R 2  , where B is a fuzzy maximal ideal of R1 or A = R1   C, where C is a fuzzy maximal ideal of R2. Proof. If A is a fuzzy maximal ideal of R. Since A is a fuzzy ideal of R, so by lemma 4.2, A = B  C for some fuzzy ideals B and C of R1, R2 respectively. Hence A = (B  C) = B  C by lemma 4.3. Then by (Lemma 2.1.(3)), B  C is a maximal ideal. So either B  C = R1  C or B  C = B  R2. hat is either B = R1 or C = R2. If B = R1, then B = R1  . If C = R2, then C = R 2  . Hence either B  C = R1  C or B  C = B  R 2  . Conversely; If A = B  R 2  and B is a fuzzy maximal ideal of R1. To prove A is a fuzzy maximal ideal of R. By (Lemma 4.3), A = B  ( R 2  ) and so A = B  R2. Since B is a fuzzy maximal ideal of R1, then by (Lemma 2.1(3)), B is a maximal ideal of R1. Hence B  R2 = A is a maximal ideal of R. On the other hand, A(0,0) = min{B(0), R 2  (0)}. But B(0) = 1 by (Lemma 2.1(1)), so A(0,0) = min{1,1} = 1. Then by (Lemma.2.1(5)), A is a fuzzy maximal ideal of R. Similarly, if A = R1   C, C is a fuzzy maximal ideal of R2, then A is a fuzzy maximal ideal of R. Now, we can give the main results, first we have the following: IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 Theorem 4.5: Let R1, R2 be two rings, let R = R1  R2 and A, B be fuzzy ideals of R1, R2 respectively. Then (1) A is a fuzzy semimaximal ideal of R if and only if A  R 2  is a fuzzy semimaximal ideal of R. (2) B is a fuzzy semimaximal ideal of R2 if and only if R1   B is a fuzzy semimaximal ideal of R. Proof (1). Since A is a fuzzy semimaximal ideal of R1, A = 1  n i i , where Ai is a fuzzy maximal ideal of R1, for all i = 1, 2, , n. Hence, A  R2 = 1  n i i  R 2  = 1  n i i  R2 ( ) n times  , since R 2 = R R2 2    n times   = R2 1 ( )    n i i  , by Lemma 2.4 (16) But by lemma 4.4, Ai  R 2  is a fuzzy maximal ideal of R, for all i = 1, 2, , n. Thus A  R 2  is a fuzzy semimaximal ideal. Conversely; If A  R 2  is a fuzzy semimaximal ideal of R, then A  R 2  = 1 D  n i i , where Di is a fuzzy maximal ideal of R for all i = 1, 2, , n. By lemma 4.2, for each i = 1, 2, , n, Di = Bi  Ci, where Bi is a fuzzy ideal of R1, Ci is a fuzzy ideal of R2. Hence A  R 2  = 1 ( C )    n i i i = 1  n i i  1 C  n i i , by Lemma2.4 (16) It follows that A = 1  n i i , 1 C  n i i = R 2  . But 1 C  n i i = R 2  implies that Ci = R 2  ,  i = 1, 2, , n. Hence Bi  Ci = Bi  R 2  ; That is Di = Bi  R 2  . Then by lemma 4.4, Bi is a fuzzy maximal ideal of R1 and so A = 1  n i i is a fuzzy semimaximal ideal of R1. (2). The proof is similarly. Next, we can give the following: IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 Theorem 4.6: Let R1, R2 be two rings, let R = R1  R2 and let A be a fuzzy ideal of R. If A is a fuzzy semimaximal of R, then either: (1) There exists fuzzy semimaximal ideals B, C of R1, R2 respectively such that A = B  C, or (2) There exists a fuzzy semimaximal ideal B of R1 such that A = B  R 2  , or (3) There exists a fuzzy semimaximal ideal C of R2 such that A = R1   C. Proof. If A is a fuzzy semimaximal ideal of R, then A = 1  n i i , where Ai is a fuzzy maximal ideal of R. By lemma 4.4, for each i = 1, 2, , n, either Ai = Bi  R 2  or Ai = R 2   Ci, where Bi, Ci are fuzzy maximal ideals of R1, R2 respectively. If Ai = Bi  R 2  , for all i = 1, 2, , n, then A = 1  n i i = 1  n i i  R 2  , putting 1  n i i = C, we get A = B  R 2  and B is a fuzzy semimaximal ideal of R1. If Ai = R1   Ci, for all i = 1, 2, , n, then A = R1   C, where C = 1 C  n i i and C is a fuzzy semimaximal ideal of R2. Now if Ai = Bi  R 2  , for some i = 1, 2, , n. Then without loss of generality, we can assume that Ai = Bi  R 2  , for some i = 1, 2,, k, k < n and Ai= R1  Ci, for all i = k+1, , n. Hence 1  n i i = 1  k i i  1        n i i k 1  n i i = R2 1 ( )    k i i  1 1 ( C )    n R i i k  = R 2 1        k i i   1 1 C         n R i i k  , by (Lemma 2.4(16)) = 1  k i i  1 C   n i i k Letting B = 1  k i i , C = 1 C   n i i k , then B, C are fuzzy semimaximal of R1, R2 respectively. Thus A = B  C. Remark 4.7: The converse of theorem 4.6 is not necessary true in general, in fact when B, C are fuzzy semimaximal ideals of R1  R2, then B  C = A need not a fuzzy semimaximal ideal of R1  R2. We can give the following example: Example: Let B, C :Z  [0,1] defined by IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 1 2 , ( ) 1 otherwise 3         x x 1 3 , C( ) 1 otherwise 2        x x B, C are fuzzy semimaximal ideals of Z since B, C are fuzzy maximal ideals of Z. On the other hand, B  C: Z  Z  [0,1] and 1 ( , ) 2 3 , 1 ( C)( ) ( , ) 2 ( 3 ), 2 1 otherwise 3                     a b a b a b B  C is not a fuzzy semimaximal ideal, since Im (BC) = 3. References 1. Malik, D.S. and Mordeson, J.N., (1991), "Fuzzy Maximal, Radical and Primary Ideals of a Ring", Information Sciences, 53: 237-250. 2. Goodreal, K.R., (1979), "Ring Theorey-Non Singular Ring and Modules", Marcei- Dekker, New York and Basel. 3. K.Y.Hatem, (2007), "Semimaximal Submodules", Ph.D.Thesis, University of Baghdad. 4. Zaheb, L.A. (1965), "Fuzzy Sets, Information and Control", 8:338-353. 5. Zahedi, M.M. (1992), "On L-Fuzzy Residual Quotient Modules and P.Primary Submodules", Fuzzy Sets and Systems,51:333-344. 6. Zahedi, M.M, (1991), "A Characterization of L-Fuzzy Prime Ideals", Fuzzy Sets and Systems, Vol.44, pp.147-160. 7. Kumar, R. (1991), "Fuzzy Semiprimary Ideals of Ring", Fuzzy Sets and Systems, 42:263-272. 8. Zhao Jiandi, Shik. Yue M., (1993), "Fuzzy Modules Over Fuzzy Rings", The J. of Fuzzy Math. 3: 531-540. 9. Kumar, R., (1992), "Fuzzy Cosets and Some Fuzzy Radicals", Fuzzy Sets and Systems, Vol.46, pp.261-265. 10. Liu, W.J, (1982), "Fuzzy Invariant Subgroups and Fuzzy Ideals", Fuzzy Sets and Systems, 8:133-139. 11. Mordeson, J.N. (1996), "Fuzzy Intersection Equations and Primary Representations", Fuzzy Sets and Systems, 83: 93-98. IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 12. Swamy, K.L.N. and Swamy, V.M., (1988), "Fuzzy Prime Ideals of Rings, J.Math. Anal.Appl, 134:94-103. 13. Hadi, I.M.A., (2001), "On Fuzzy Ideals of Fuzzy Rings", Math. and Physics J. 16(4): 14. Abu-Dareb, A.T.H., (2000), "On Qusi-Frobenius Fuzzy Rings", M.Sc. Thesis, University of Baghdad. 15. Larsen, M.D. and Carthy, P.J.Mc, (1971), "Multiplicative Theory of Ideals", Academic Press, New York. 16. Hadi, I.M.A. and Abu-Dareb, A.T.H., (2004), "P-F Fuzzy Rings and Normal Fuzzy Ring", Ibn-Al-Haitham J. of Pure and Applies Sciences, 17(1): 2002( 2) 22مجلة ابن الهيثم للعلوم الصرفة والتطبيقية المجلد المـثاليات الضبابية شبه األعظمية أنعام محمد علي هادي، ميسون عبد هامل ابن الهيثم، جامعة بغداد -قسم الرياضيات، كلية التربية الخالصة تقاطع عدد منته من مثاليات Iيسمى شبه أعظمي إذا كان Rمثالي فعلي في Iالية ذا محايد.حلقة ابد Rلتكن Rعلى Aيكون المثالي الضبابي اذ، Rعظمى. في هذا البحث قمنا بتنصيب هذا المفهوم إلى المثاليات الضبابية على فضال عن مثاليا شبه أعظمي إذا كان تقاطع عدد منته من المثاليات الضبابية العظمى. خواص أساسية مختلفة قد أعطيت قد أعطيت بعض األمثلة لتوضيح هذا المفهوم. هذا