IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 Semiessential Fuzzy Ideals and Semiuniform Fuzzy Rings M. A.Hamil Departme nt of Mathematics ,College of Education - Ibn-Al-Haitham ,Unive rsity of Baghdad Abstract In this p aper, we introduce and st udy semiessential fuzz y ideals of fuz zy rings, uniform fuzzy rings and semiuniform fuzzy rings. Introduction Zadah in [1] introduced the notion of a fuzz y subset A of a nonempty set S as a mapp ing from S into [0,1], Liu in [2] introduced the concept of a fuzzy ring, M artines [3] introduced the notion of a fuzz y ideal of a fuzz y ring. A non zero p rop er 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 p rop erties. Nada in [6] introduced and st udied 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 p rime ideals of R, [4]. A ring R is called uniform if every ideal of R is essential. Nada in [6] introduced and st udied the notion semiuniform ring where a ring R is called semiuniform ring if every ideal of R is semiessential ideal. In this p aper 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 p rime fuzz y ideal P of X. A fuzzy ring X is called uniform (semiuniform) if every fuzzy ideal of X is essential (semiessential) resp ectively. In S.1, some basic definitions and results are collected. In S.2, we st udy semiesential fuzzy ideals of fuzzy ring, we give some basic prop erties about t his concept. In S.3, we st udy the notion of uniform fuzzy rings and semiuniform fuzzy rings. Several p rop erties about t hem are given. Throughout this p aper, 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 fuz zy 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 fuzz y 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 map p ing from a set M into a set N, let B be a fuzz y 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. S CI. VOL.22 (4) 2009 If f is a function from a set M into a set N, let A1 and A2 be fuzz y subsets of M and B1, B2 be fuzzy subsets of N, t hen 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]. M oreover the following definitions and p rop erties are needed later 1.1 De fini tion [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 De fini tion [3] Let X be a fuzz y ring of R, a fuzz y 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. Not e that if X is a fuzz y ring of R, then X(a)  X(0),  a  R, [3, p rop osition 2.7] If A is a fuzzy ideal of X, t hen A(a)  A(0),  a  R, [3, p rop osition 2.9]. 1.3 Proposi tion A fuzzy subset X : R  [0,1], is a fuzz y ring if Xt is a subring of R,  t  [0,X(0)], (3, p rop osition 2.10 (i)). Given a fuzzy ring X and a fuzzy set A : R  [0,1] with A  O1, then A is a fuzz y ideal of X iff At is an ideal of Xt  t  [0,A(0)], [3, p rop osition 2.10 (iii)]. 1.4 De fini tion [9] A fuzzy ideal P of a fuzzy ring X is called a p rime fuzzy ideal if P  R (where R denotes t he 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 De fini tion [10] Let X and Y be fuzzy rings of R1 and R2 resp ectively. 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 resp ectively, then A  B : R1R2  [0,1] defined by : (A  B)(a,b) = min{A(a),B(b)},  (a,b)  R1R2. 1.6 Proposi tion [9] Let X , Y be two fuzzy rings of R and R resp ectively and f : R R is a homomorp hism, then If A is a p rime fuzz y ideal of X and A is f-invariant, then f (A) is a p rime fuzz y ideal of Y. 1.7 De fini tion [10] Let A be a fuzz y ideal of a fuzz y ring X of a ring R, A is called an essential fuzzy ideal if A  K  O1 for each fuzz y ideal K of X, K  O1. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 S.2 Semie ssential 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 p rop erties of this concept. 2.1 De fini tion 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 fuzz y ideal of X if ABO1, for any p rime fuzz y 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 fuzz y ideal of X, t hen A is a semiessential fuzz y ideal of X. Proof: It is easy , so it is omitted. The converse of remark 2.2 is not true in general. However an examp le which will exp lain this dep endens on t heorem 2.3, so we shall give it later (see remark 2.5). 2.3 The orem Let X be a fuzzy ring of R, let A be a fuzz y 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 p rime fuzzy ideal of X such that BO1. To p rove A  B O1, since B is a p rime fuzzy ideal. Hence Bt is a p rime ideal of Xt,  t  (0,X(0)] by [11,p rop osition 1.2.9]. Which imp lies At  Bt  (0) and At  Bt = (A  B)t  (0). Hence A  B O1. Thus A is a semiessential fuzz y ideal of X. The following remark shows that t he converse of this t heorem 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 At is a semiessential ideal of Xt,  t  [0,1]. As the following examp le shows: Example: Let R = Z 6, 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 t he following examp le 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. S CI. VOL.22 (4) 2009 1 if (6) ( ) 0 otherwise.       x x It is clear A is a fuzzy ideal of a fuzz y ring X, Xt = Z 12 and At = ( 6 ),  t > 0 is a semiessential ideal in Z12 since ( 6 )  ( 3 ) = ( 6 ) and ( 6 )  ( 2 ) = ( 6 ), ( 3 ) and ( 2 ) are p rime 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 fuz zy 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 fuzz y ideal of X, t hen 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 fuzz y ideal of X, as t he following examp le 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 p rime ideals of Z12 = Xt,  t > 0. Thus B is a semiessential fuzz y ideal of X. Let 1 if (3) C( ) 0 otherwise.      x x , C is a p rime fuzzy ideal of X, since Ct = ( 3 ) is a p rime 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 fuzz y ideal of X. We can give the following examp le : 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. S CI. 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 Proposi tion Let A and B be fuzz y ideals of fuz zy ring X of a ring R such that A is an essential fuzzy ideal and B is a semiessential fuzz y ideal. Then A  B is a semiessential fuzzy ideal of X. Proof: Let P be a non-zero p rime fuzz y ideal of X, since B is a semiessential fuzz y ideal of X, then B  P ≠ O1.Also A is an essential fuzzy ideal of X, we get (A  B)  P  O1. Which imp lies A  B is a semiessential fuzzy ideal of X. 2.11 Proposi tion 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 Proposi tion 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 st udy ing the direct sum of semiessential fuzzy ideals, we need the following lemma. 2.13 Lemma Let X and Y be fuzz y rings of rings R1, R2 resp ectively. Let W be a fuzz y ideal of T = X  Y, then W is a p rime fuzzy ideal of T if there exists A and B p rime fuzzy ideals of X, Y resp ectively such that W = A  Y or W = X  B. Proof: If W is a p rime fuzzy ideal in T = X  Y. Since W is a fuzzy ideal in XY, there exists fuz zy ideal A and B of X, Y resp ectively such that W = A  B by (10,theorem 2.4.1.9). Thus Wt = At  Bt,  t  (0,1]. But W is a p rime so Wt is p rime in Tt = Xt  Yt ,  t  (0,1]. Hence either Wt = I  Yt or Wt = Xt  J where I, J are prime ideals in Xt , Yt resp ectively, by (12, p age 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 p rime fuzzy ideals of X, Y resp ectively. If W = A  Y, then Wt = (A  Y)t = At  Yt but At is a prime ideal in Xt,  t  (0,1] by (11,p rop osition 1.2.9). Hence At  Yt is p rime in Tt by (12,p age 53). That is Wt is a p rime ideal in (X  Y)t = Tt . Thus W is a p rime fuzzy ideal of X  Y = T by (11, p rop osition 1.2.9). Now we can give the following main result. 2.14 The orem Let X and Y be fuzzy rings of R1, R2 resp ectively. If A and B are semiessential fuzzy ideals of X, Y resp ectively. Then AB is a semiessential fuzzy ideal of X  Y. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 Proof: To p rove 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 p rime fuzz y ideal of X  Y, hence either W = C  Y or W = X  D, for some prime fuzz y ideals C, D of X, Y resp ectively. 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, t hen (A B) WO1. Therefore, A  B is semiessential. The converse of theorem 2.14 is not true in general as the following examp le 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 t he fuzz y rings X and Y resp ectively. 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 p rime ideals in Z6  Z12 are: < 2 >  Z12, < 3 >  Z12, Z6  < 2 >, Z 6  < 3 >. Hence ((3 )  ( 3 ))(( 2 )  Z12)0, (( 3 )  ( 3 ))  (( 3 )  Z12)  0, (( 3 )  ( 3 ))  (Z6 ( 2 ))  0 and ((3 )  ( 3 ))  (Z6 ( 3 ))  0. Which imp lies (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 fuzz y ideal of X, since there exists p rime 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 fuzz y ideal of Y. Next, we have the following p rop osition about the inverse image of semiessential fuzz y ideals. 2.16 Proposi tion Let X and Y be fuzzy rings of rings R1, R2 resp ectively. Let f : R1  R2 be a homomorp hism. If A is a semiessential fuzzy ideal of Y, then f - 1 (A) is a semiessential fuzz y ideal of X. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 Proof: By (3,p rop osition 3.3), f - 1 (A) is a fuzzy ideal of X. Let B be a prime fuzz y ideal of X and B is f-invariant. To p rove 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 p rime fuzzy ideal of Y by p rop osition 1.6. Therefore, A  f (B)  OY(0), since A is semiessential. On t he 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 fuzz y ideal of X. S .3 Uni form and S emiuniform Fuz zy 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 st udy the notion of uniform and semiuniform fuzzy rings and give many p rop erties about t hem. 3.1 De fini tion 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) fuzz y ideal of X. 3.2 Proposi tion Let X be a uniform fuzz y 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 examp le 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, At is an ideal of Xt,  t [0,1] and Xt=Z 36. 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 fuzz y 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, p rop osition 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. T o p rove A is a semiessential in X. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 Sup p ose there exists a non-zero p rime fuzz y ideal P of X such that A  P = O1 since P  O1, there exist t1  (0,1] such that 1 P t  {0}. On the other hand, (A P) = O1. Implies 1 1 11 P (O ) {0}    t t t . But this is a contradiction since 1 P t  {0}, 1 P t is a p rime in Xt , (11, p rop osition 1.2.9) and 1  t is semiessential in Xt. Hence AP  O1 and A is a semiessential fuzz y 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 excep t the direct sum of essential fuzzy ideals. Re ferences 1. Zadah, L.A., (1965), "Fuz zy Sets", Inform and Control, 8, 338-353. 2. Liu, W.J., (1982), "Fuz zy Invariant Subgroup and Fuzzy Ideals", Fuz zy Sets and Sy st ems, 8, 133-139. 3. M artines, L., (1995), "Fuzzy Subgroup s of Fuzzy Group s and Ideals of Fuz zy Rings", The Journal of Fuzzy M ath., 3, No. 4, 833-849. 4. Kasch, F., (1982), "M odules and Rings", Academic Press, London, New York. 5. Hadi, M .A. Inaam, (2001), "Some Sp ecial Fuz zy Ideals of Fuz zy Rings', J. M ath. And Phy sic, 6, No.2. 6. Al-Daban, K.A.Nada, (2005), "Semiessential Submodules and Semiuniform M odules", M .Sc. Thesis, T ukirt University . 7. Al-Khamees, Y. and M ordeson, (1998), "Fuz zy Princip al Ideals and Fuz zy Simp le Field Extensions", Fuz zy Sets and Sy st ems, 96, 147-253. 8. Kumar, R., (1991), "Fuz zy Semiprimary Ideals of Ring", Fuz zy Sets and Sy st ems, 42, 263-272. 9. M artines, L., (1999), "Prime and Primary L-Fuzzy Ideals of L-Fuzzy Rings", Fuzzy Sets and Sy st ems, 101, 489-494. 10. Abo-Drab, A.T ., (2000), "Almost Quasi-Forbenius Fuz zy Rings", M .Sc. Thesis, University of Baghdad, College of Education, Ibn-AlHaitham. 11. M egeed, N.R., (2000), "Some Results on Ctegories of Rings", Fuz zy Ring and Its Sp ectrum, M .Sc. Thesis, University of Baghdad, College of Education, Ibn-AlHaitham. 12. Larsen, M .D. and M c.Carthy P.J., (1971), "M ultip licative Theory of Ideals", Academic Press, New York. (22مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة المجلد 4 (2009 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