IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.22 (4) 2009 Semiessential Fuzzy Ideals and Semiuniform Fuzzy Rings M. A.Hamil Department of Mathematics ,College of Education - Ibn-Al-Haitham ,University of Baghdad Abstract In this paper, we introduce and study semiessential fuzzy ideals of fuzzy rings, uniform fuzzy rings and semiuniform fuzzy rings. Introduction Zadah in [1] introduced the notion of a fuzzy subset A of a nonempty set S as a mapping from S into [0,1], Liu in [2] introduced the concept of a fuzzy ring, Martines [3] introduced the notion of a fuzzy ideal of a fuzzy ring. A non zero proper ideal I of a ring R is called an essential ideal if I  J  (0), for any non zero ideal J of R, [4]. Inaam in [5] fuzzified this concept to essential fuzzy ideal of fuzzy ring and gave its basic properties. Nada in [6] introduced and studied notion of semiessential ideal in a ring R, where a non zero ideal I of R is called semiessential if I  P  (0) for all non zero prime ideals of R, [4]. A ring R is called uniform if every ideal of R is essential. Nada in [6] introduced and studied the notion semiuniform ring where a ring R is called semiuniform ring if every ideal of R is semiessential ideal. In this paper we fuzzify the concepts semiessential ideal of a ring, uniform ring and semiuniform ring into semiessential fuzzy ideal of fuzzy ring, uniform fuzzy ring and semiuniform fuzzy ring. Where a fuzzy ideal A of a fuzzy ring X is semiessential if I  P  (0) for any prime fuzzy ideal P of X. A fuzzy ring X is called uniform (semiuniform) if every fuzzy ideal of X is essential (semiessential) respectively. In S.1, some basic definitions and results are collected. In S.2, we study semiesential fuzzy ideals of fuzzy ring, we give some basic properties about this concept. In S.3, we study the notion of uniform fuzzy rings and semiuniform fuzzy rings. Several properties about them are given. Throughout this paper, R is commutative ring with unity , and X(0) = 1, for any fuzzy ring. S.1 Preliminaries Let R be a commutative ring with identity . A fuzzy subset of R is a function from R into [0,1]. Let A and B be a fuzzy subsets of R we write A  B if A(x)  B(x), for all x  R, (1) and (A  B)(x) = min {A(x), B(x)},  x  R. For each t  [0,1], the set { x  R; A(x)  t} is called the level subset of R, [7]. If A and B are fuzzy subsets of R, then  t  [0,1] 1. (A  B)t = At  Bt, [1], 2. A = B iff At = Bt, [1] . Let f be a mapping from a set M into a set N, let B be a fuzzy subset of N. The inverse image of B is a fuzzy subset of M defined by f – 1 (B)(x) = B(f (x)),  x M, [1]. Let A be a fuzzy subset of a set M. A is called an f-invariant if A(x) = A(y), whenever f (x) = f (y), where x, y M, [8]. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.22 (4) 2009 If f is a function from a set M into a set N, let A1 and A2 be fuzzy subsets of M and B1, B2 be fuzzy subsets of N, then 1. f (A1  A2) = f (A1)  f (A2), A1, A2 are f-invariant, [8] 2. f – 1 (B1  B2) = f – 1 (B1)  f – 1 (B2), [8]. 3. f – 1 (f (A1)) = A1, whenever A1 is f-invariant, [8] 4. f ( f – 1 (B1)) = B1, [8]. Moreover the following definitions and properties are needed later 1.1 Definition [2] Let X be a fuzzy subset of a ring R. X is called a fuzzy ring of R if X  O1 and for each x, y  R 1. X(x – y)  min {X(x),X(y)}, 2. X(x y)  max {X(x),X(y)}. 1.2 Definition [3] Let X be a fuzzy ring of R, a fuzzy subset A of R is called a fuzzy ideal of X if 1. A  X, 2. A(x – y)  min {A(x), A(y)},  x, y  R, 3. A(x y)  max {A(x), X(y)},  x, y  R. Note that if X is a fuzzy ring of R, then X(a)  X(0),  a  R, [3, proposition 2.7] If A is a fuzzy ideal of X, then A(a)  A(0),  a  R, [3, proposition 2.9]. 1.3 Proposition A fuzzy subset X : R  [0,1], is a fuzzy ring if Xt is a subring of R,  t  [0,X(0)], (3, proposition 2.10 (i)). Given a fuzzy ring X and a fuzzy set A : R  [0,1] with A  O1, then A is a fuzzy ideal of X iff At is an ideal of Xt  t  [0,A(0)], [3, p roposition 2.10 (iii)]. 1.4 Definition [9] A fuzzy ideal P of a fuzzy ring X is called a prime fuzzy ideal if P  R (where R denotes the characteristic function of R such that R(x) = 1,  x  R) and it satisfies: min {P(x y),X(x),X(y)}  max {P(x),P(y)} for all x, y R. IfX = R, then P(x y)  max {P(x),P(y)} for all x, y  R. 1.5 Definition [10] Let X and Y be fuzzy rings of R1 and R2 respectively. Then the direct sum of X and Y (denoted by XY) is defined by: XY : R1R2  [0,1] such that (XY)(a,b) = min{X(a),Y(b)},  (a,b)  R1R2. If A, B are fuzzy ideals of X and Y respectively, then A  B : R1R2  [0,1] defined by : (A  B)(a,b) = min{A(a),B(b)},  (a,b)  R1R2. 1.6 Proposition [9] Let X , Y be two fuzzy rings of R and R respectively and f : RR is a homomorphism, then If A is a prime fuzzy ideal of X and A is f-invariant, then f (A) is a prime fuzzy ideal of Y. 1.7 Definition [10] Let A be a fuzzy ideal of a fuzzy ring X of a ring R, A is called an essential fuzzy ideal if A  K  O1 for each fuzzy ideal K of X, K  O1. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.22 (4) 2009 S.2 Semiessential Fuzzy Ideals In this section, we introduce the notion of semiessential fuzzy ideals of fuzzy ring as a generalization of (ordinary) notion semiessential ideals of a ring. We shall give many properties of this concept. 2.1 Definition Let X be a fuzzy ring of a ring R. Let A be a fuzzy ideal of X such that A OX(0)=O1. A is called a semiessential fuzzy ideal of X if ABO1, for any prime fuzzy ideal B of X. 2.2 Remark Let X be a fuzzy ring of a ring R. It is clear that if A is an essential fuzzy ideal of X, then A is a semiessential fuzzy ideal of X. Proof: It is easy , so it is omitted. The converse of remark 2.2 is not true in general. However an example which will explain this dependens on theorem 2.3, so we shall give it later (see remark 2.5). 2.3 Theorem Let X be a fuzzy ring of R, let A be a fuzzy ideal of X, if At is a semiessential ideal of Xt,  t  (0,1]. Then A is a semiessential fuzzy ideal of X. Proof: Let B be a prime fuzzy ideal of X such that BO1. To prove A  B O1, since B is a prime fuzzy ideal. Hence Bt is a prime ideal of Xt,  t  (0,X(0)] by [11,proposition 1.2.9]. Which implies At  Bt  (0) and At  Bt = (A  B)t  (0). Hence A  B O1. Thus A is a semiessential fuzzy ideal of X. The following remark shows that the converse of this theorem is not true in general. 2.4 Remark If X is a fuzzy ring of a ring R, A is a semiessential fuzzy ideal of X, then it is not necessarily that A t is a semiessential ideal of Xt,  t  [0,1]. As the following example shows: Example: Let R = Z6, define X: Z6  [0,1], A: Z6  [0,1] by: 1 if 0 1 X( ) if 2,4 2 0 otherwise.       a a a 1 if 0 1 ( ) if 2,4 3 0 otherwise.        a a a It is clear X is a fuzzy ring of Z6, A is a fuzzy ideal of X and A  O1. A is an essential fuzzy ideal of X see (5, remark 2.3). Hence A is semiessential in X. On the other hand, 1 2  = {0}, 1 2 X = {0,2,4}. Hence A 1 2 is not semiessential ideal in X 1 2 . 2.5 Remark If X is a fuzzy ring of a ring R, A is a semiessential fuzzy ideal of X, then it is not necessarily that A is an essential fuzzy ideal of X, as the following example shows: Example: Let R = Z12, define X: Z12  [0,1] by X(a) = 1,  a  Z12, let A: Z12  [0,1] define by: IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.22 (4) 2009 1 if (6) ( ) 0 otherwise.       x x It is clear A is a fuzzy ideal of a fuzzy ring X, X t = Z12 and At = ( 6),  t > 0 is a semiessential ideal in Z12 since ( 6)  (3 ) = ( 6) and ( 6)  ( 2 ) = ( 6), (3 ) and ( 2 ) are prime ideals of Z12. Thus A is a semiessential fuzzy ideal by Theorem 2.3. But A is not an essential fuzzy ideal since there exists fuzzy ideal B of X defined by : 1 if (4) ( ) 0 otherwise.       x x B  O1.But A  B = O1. 2.6 Remark Let X be a fuzzy ring of R, let A and B be fuzzy ideals of X such that A  B. If A is a semiessential. Then B is a semiessential fuzzy ideal of X. Proof: It is clear. 2.7 Corollary If A and B are fuzzy ideals of a fuzzy ring X of a ring R such that A  B is a semiessential fuzzy ideal of X, then A and B are semiessential fuzzy ideals of X. 2.8 Remark Let A and B be fuzzy ideals of fuzzy ring X of a ring R such that A  B and B is a semiessential, then it not necessarily that A is semiessential fuzzy ideal of X, as the following example shows: Example: Let X: Z12  [0,1], A: Z12  [0,1], B: Z12  [0,1] defined by: X(a)= 1,  a  Z12, 1 if (4) ( ) 0 otherwise. x x       and 1 if (2) ( ) 0 otherwise. x x       . It is clear that Xt = Z12 ( t > 0) and A, B are fuzzy ideals of fuzzy ring X, Bt = ( 2 ) is a semiessential ideal of Xt,  t > 0, since ( 2 )  ( 2 ) = ( 2 ) and ( 2 )  (3 ) = ( 6) where ( 2 ) and (3 ) are the only prime ideals of Z12 = Xt,  t > 0. Thus B is a semiessential fuzzy ideal of X. Let 1 if (3) C( ) 0 otherwise.      x x , C is a prime fuzzy ideal of X, since Ct = (3 ) is a prime ideal of Xt,  t > 0. But A  C= O1. Thus A is not a semiessential fuzzy ideal of X. 2.9 Remark If A and B are semiessential fuzzy ideals of fuzzy ring X of a ring R. Then it not necessarily that A  B is semiessential fuzzy ideal of X. We can give the following example : Example: Let X: Z36  [0,1] define by X(a) = 1,  a  Z36 and let A: Z36  [0,1], B: Z36  [0,1] defined by: 1 if (12) ( ) 0 otherwise.       x x 1 if (18) ( ) 0 otherwise.       x x IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.22 (4) 2009 A and B are fuzzy ideals of X. But for each t  (0,1], At = (12 ), Bt = (18 ). It is easy to show that At and Bt are semiessential ideals of Xt = Z36. Thus A and B are semiessential fuzzy ideals of X by Theorem 2.3. But A  B = O1. Thus A  B is not a semiessential fuzzy ideal of X. 2.10 Proposition Let A and B be fuzzy ideals of fuzzy ring X of a ring R such that A is an essential fuzzy ideal and B is a semiessential fuzzy ideal. Then A  B is a semiessential fuzzy ideal of X. Proof: Let P be a non-zero prime fuzzy ideal of X, since B is a semiessential fuzzy ideal of X, then B  P ≠ O1.Also A is an essential fuzzy ideal of X, we get (A  B)  P  O1. Which implies A  B is a semiessential fuzzy ideal of X. 2.11 Proposition Let X. be a fuzzy ring of R such that X(a) = 1,  a  R. Let I be a semiessential ideal of R. If A : R  [0,1] defined by: 1 if I ( ) r if a I      a a where r  (0,1). Then A is a semiessential fuzzy ideal of X. Proof: It is easy , so it omitted. 2.12 Proposition Let X. be a fuzzy ring of R such that X(a) = 1,  a  R. Let I be a ideal of R. Then I is a semiessential ideal of R if I is a semiessential fuzzy ideal of X where I 1 if I ( ) 0 otherwise      x x Proof: It is easy , so it is omitted. Before studying the direct sum of semiessential fuzzy ideals, we need the following lemma. 2.13 Lemma Let X and Y be fuzzy rings of rings R1, R2 respectively. Let W be a fuzzy ideal of T = X  Y, then W is a prime fuzzy ideal of T if there exists A and B prime fuzzy ideals of X, Y respectively such that W = A  Y or W = X  B. Proof: If W is a prime fuzzy ideal in T = X  Y. Since W is a fuzzy ideal in XY, there exists fuzzy ideal A and B of X, Y respectively such that W = A  B by (10,theorem 2.4.1.9). Thus Wt = At  Bt,  t  (0,1]. But W is a prime so Wt is prime in T t = Xt  Yt ,  t  (0,1]. Hence either Wt = I  Yt or Wt = Xt  J where I, J are prime ideals in Xt , Yt respectively, by (12, page 53). Therefore I= A t or J = Bt and hence Wt = At  Yt or Wt = Xt  Bt. It follows that Wt = (A  Y)t or Wt =(X B)t. Thus W = A  Y or W = X  B. Conversely; If W = A  Y or W = X  B, where A and B are prime fuzzy ideals of X, Y respectively. If W = A  Y, then Wt = (A  Y)t = At  Yt but At is a prime ideal in Xt,  t  (0,1] by (11,proposition 1.2.9). Hence At  Yt is prime in T t by (12,page 53). That is Wt is a prime ideal in (X  Y)t = T t . Thus W is a prime fuzzy ideal of X  Y = T by (11, proposition 1.2.9). Now we can give the following main result. 2.14 Theorem Let X and Y be fuzzy rings of R1, R2 respectively. If A and B are semiessential fuzzy ideals of X, Y respectively. Then AB is a semiessential fuzzy ideal of X  Y. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.22 (4) 2009 Proof: To prove A  B is semiessential fuzzy ideal of X  Y. Since A  B is a non-zero fuzzy ideal of XY by (10, Theorem 2.4.1.9), there exists (a,b)  R1  R2 such that (A  B)(a,b) = min{A(a),B(b)}  0. Thus A(a)  0 and B(b)}  0. Now, let W be a non-zero prime fuzzy ideal of X  Y, hence either W = C  Y or W = X  D, for some prime fuzzy ideals C, D of X, Y respectively. Assume W = C  Y. If C = O1, then (A  B)  W = (A  B)  (C  Y) = (A  C)  (B  Y) = O1  B But (O1  B)(a,b) = min{O1(0),B(b)} = min {1, B(b)} = B(b)  0 Thus (A  B)  W  O1. If C  O1, then A  C  O1, since A is semiessential in X. Hence there exists a1  R1 such that (A  C)(a1) O1, so min{A(a1),C(a1)}0, since (AB)(CY)= (A  C)  (B  Y) = (A  C)  B. It follows that [(A  C)  B](a,b) = min{A(a1),C(a1),B(b)}  0. That [(A  B)  (C  Y)  O(X  Y)  O1(0,0). Similarly, if W = X  D, then (A B) WO1. Therefore, A  B is semiessential. The converse of theorem 2.14 is not true in general as the following example shows; 2.15 Example Let X: Z6  [0,1], Y: Z12  [0,1] define by X(a) = 1,  a  Z6, Y(b) =1,  b  Z12, let A: Z6  [0,1], B: Z12  [0,1] defined by: 1 if {0,3} ( ) 0 otherwise. x x       ,  x  Z6 1 if {0,3,6,9} ( ) 0 otherwise. x x       ,  x  Z12 It is easy to check that A and B are fuzzy ideals of the fuzzy rings X and Y respectively. 1 if {0,3}, {0,3,6,9} ( )( , ) 0 otherwise.        x y x y (A  B)t = A t  Bt = <3 >  <3 > is a semiessential ideal in Z6  Z12. Since the prime ideals in Z6  Z12 are: < 2 >  Z12, <3 >  Z12, Z6  < 2 >, Z6  <3 >. Hence ((3 )  (3 ))(( 2 )  Z12)0, ((3 )  (3 ))  ((3 )  Z12)  0, ((3 )  (3 ))  (Z6 ( 2 ))  0 and ((3 )  (3 ))  (Z6 (3 ))  0. Which implies (A  B)t is semiessential in (X  Y)t,  t. Thus A  B is semiessential fuzzy ideal of X  Y by Theorem 2.3. But A is not semiessential fuzzy ideal of X, since there exists prime fuzzy ideal C of X such that A  C = O1, where 1 if {0,2,4} C( ) 0 otherwise.      x x  x  Z6. Similarly, we can show that B is not semiessential fuzzy ideal of Y. Next, we have the following proposition about the inverse image of semiessential fuzzy ideals. 2.16 Proposition Let X and Y be fuzzy rings of rings R1, R2 respectively. Let f : R1  R2 be a homomorphism. If A is a semiessential fuzzy ideal of Y, then f - 1(A) is a semiessential fuzzy ideal of X. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.22 (4) 2009 Proof: By (3,proposition 3.3), f - 1 (A) is a fuzzy ideal of X. Let B be a prime fuzzy ideal of X and B is f-invariant. To prove that f - 1 (A)  B  OX(0) = O1. f (f - 1(A)  B) = f (f - 1(A)  f (B)) = A  f (B) since A, B are f-invariant. However f (B) is prime fuzzy ideal of Y by proposition 1.6. Therefore, A  f (B)  OY(0), since A is semiessential. On the other hand, f - 1(A  f (B)) = f - 1(A)  f – 1(f (B)) = f - 1 (A)  B, since B f-invariant  OX(0)  O1. Thus f - 1(A) is a semiessential fuzzy ideal of X. S.3 Uniform and Semiuniform Fuzzy Rings Recall that a ring R is called a uniform ring if every non zero ideal I of R is an essential ideal, [4] and a ring R is called semiuniform if every non zero ideal I of R is a semiessential ideal of R, [6]. In this section, we introduce and study the notion of uniform and semiuniform fuzzy rings and give many properties about them. 3.1 Definition Let X be a fuzzy ring of a ring R. X is called uniform (semiuniform) if every non zero fuzzy ideal A of X is an essential (semiessential) fuzzy ideal of X. 3.2 Proposition Let X be a uniform fuzzy ring, then X is semiuniform fuzzy ring. Proof: It is follows by Remark 2.2. But the converse is not true in general. 3.3 Remark If X is a semiuniform fuzzy ring. Then it is not necessarily that X is uniform fuzzy ring as the following example shows: Example: Let X: Z36  [0,1] define by X(a) = 1,  a  Z36. It is easy to check that X is a fuzzy ring of Z36. For each fuzzy ideal A of X, A t is an ideal of Xt,  t [0,1] and Xt=Z36. The ideals of Z36 are : (0), (2), (3), (4),(6),(12) and (18)which are semiessential ideals in Z36. So A is a semiessential fuzzy ideal of X by Theorem 2.3. Thus X is a semiuniform fuzzy ring. But X is not uniform fuzzy ring since there exist fuzzy ideals A and B of X such that 1 if (12) ( ) 0 otherwise.       x x 1 if (18) ( ) 0 otherwise.       x x ,  x  Z36. A is not essential fuzzy ideal since A  B = O1. Thus X is not uniform fuzzy ring. 3.4 Remark Let X be a fuzzy ring of a ring R such that Xt is a uniform ring,  t  (0,1]. Then X is a uniform fuzzy ring. Proof: It is follow by [5, proposition 2.4]. 3.5 Remark Let X be a fuzzy ring of a ring R such that Xt is a semiuniform ring,  t , then X is a semiuniform fuzzy ring. Proof: Let A be a non-zero fuzzy ideal of X. To prove A is a semiessential in X. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.22 (4) 2009 Suppose there exists a non-zero prime fuzzy ideal P of X such that A  P = O1 since P  O1, there exist t1  (0,1] such that 1 Pt  {0}. On the other hand, (A P) = O1. Implies 1 1 11P (O ) {0}   t t t . But this is a contradiction since 1 Pt  {0}, 1 Pt is a prime in Xt , (11, proposition 1.2.9) and 1 t is semiessential in Xt. Hence AP  O1 and A is a semiessential fuzzy ideal in X. Thus X is a semiuniform fuzy ring. The notion of semiessential fuzzy ideal of fuzzy ring can be generalized to semiessential fuzzy submodules similarly we can obtain similar results except the direct sum of essential fuzzy ideals. References 1. 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Abo-Drab, A.T., (2000), "Almost Quasi-Forbenius Fuzzy Rings", M.Sc. Thesis, University of Baghdad, College of Education, Ibn-AlHaitham. 11. Megeed, N.R., (2000), "Some Results on Ctegories of Rings", Fuzzy Ring and Its Spectrum, M.Sc. Thesis, University of Baghdad, College of Education, Ibn-AlHaitham. 12. Larsen, M.D. and Mc.Carthy P.J., (1971), "Multiplicative Theory of Ideals", Academic Press, New York. 2009) 4 (22مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة المجلد المثالیات شبه الجوهریة الضبابیة والحلقات شبه المنتظمة الضبابیة میسون عبد هامل ابن الهیثم ، جامعة بغداد-كلیة التربیة، قسم الریاضیات الخالصة فـي هــذا البحــث قــدمنا ودرســنا المثالیـات شــبه الجوهریــة الــضبابیة فــي حلقـة ضــبابیة، الحلقــات المنتظمــة الــضبابیة .والحلقات شبه المنتظمة الضبابیة