On Intuitionistic Fuzzy bi- Ideal With respect To an Element of a near Ring

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1 ISSN: On Intuitionisti Fuzzy bi- Ideal With respet To an Element o a near Ring Showq Mohammed.* Department o Mathemati, College o Eduation or Girls, L Kua University, l-kua, Iraq bstrat In this paper we introdue the notions o bi-ideal with respet to an element r denoted by r-bi- ideal ) o a near ring, and the notion uzzy bi- ideal with respet to an element o a near ring and the relation between F-r-bi-ideal and r-bi-ideal o the near ring, we studied the image and inverse image o r-bi- ideal under epimomorphism,the intersetion o r-bi- ideals and the relation between this ideal and the quasi ideal o a near ring, also we studied the notion intuitionisti uzzy biideal with respet to an element r o the near ring N, and give some theorem about this ideal. Keywords: Near ring, ideal o a near ring,bi- ideal, quasi- ideal, P-regular o near ring, intuitionisti uzzy, intuitionisti uzzy bi-ideal. الحدس الضبابي للمثالية bi بالنسبة لعنصر ما في الحلقة القريبة الخالصة قدمنا في هذا البحث شوق محمد اب ارهيم* قسم الرياضيات, كلية التربية للبنات, جامعة الكوفة, الكوفة, الع ارق بعض المفاهيم ومنها المثالية bi-ideal والتي يرمز لها بالرمز,)r-bi-ideal التشاكل بالنسبة لعنصر ما في الحلقة القريبة N كما ودرسنا الصور المباشرة ومعكوس الصورة للمثالية تحت الشامل كما وأعطينا بعض الخواص التي تتعلق بهذه المثالية وعالقتها ببعض المثاليات المعروفة bi-ideal بالنسبة لعنصر ما في الحلقة القريبة N وتم intuitionisti uzzy bi- ideal وأيضا تطرقنا إلى مفهوم جديد وهو المثالية الضبابية إعطاء يعض المبرهنات والنتائج الخاصة بها.وكما مفهوم قدمنا بالنسبة لعنصر ما في الحلقة القريبة مع إعطاء بعض المبرهنات واالمثلة عليها. Introdution The notion o near ring is irst deine by G. Pliz [1] in 1983, the notion o bi- ideal interused by N.Ganesan [],in 1986 the notion intutionistis uzzy sets denoted by K.T. tanassov [3], in 1987 T.T. Chelvam, N. Ganesan denoted the notion bi-ideals o near-rings, in 1997 the notion Fuzzy Ideal denoted by D.T.K,and Biswas[4], in 01 the notion P-regular near ring denoted by phisit in [5]. 1.Preliminaries In this setion we give some onepts that we need. Deinition 1.1) [1] let near ring is a set N together with two binary operations + and. suh that 1) N,+) is a group not neessarily abelian ) ) N,.) is a semigroup. 3) n 1 + n ). n 3 = n 1. n 3 + n. n 3 For all n1, n, n 3, N, Deinition 1.) [6]: Let N be a near ring. normal subgroup I o N,+) is alled a let ideal o N i * mshowq@yahoo.o.uk 1806

2 . I.N I. n, n1 N and or all ii,n1 i) n n1 I Deinition 1.3 ) [7] Let N 1,+,.) and N,+,. ) be two near rings. The mapping : N 1 N is alled a near ring homomorphism i or all m, n N 1 m+ n) = m) + n), m. n) = m). n). Theorem 1.4) [5] Let : N 1,,.) N,,. ) be a near ring homomorphism 1) I I is an ideal o a near ring N 1, then I) is an ideal o a near ring N. ) I J is an ideal o a near ring N, then -1 J) is an ideal o the near ring N 1 Deinition 1.5 ) [8] nonempty subset Q o a near ring N is alled a quasi ideal o N i 1) Q together with addition is a subgroup o N. ) QN NQ Q. Deinition 1.6 ) [] nonempty subset B o a near ring N is alled a bi-ideal o N i 1) B together with addition is a subgroup o N. ) BNB B. Deinition 1.7 ) [9] Let N be a near ring with unity and P an ideal o N.Then N is said to be P-regular near ring i or eah x N there exists y N suh that xyx x P. Deinition 1.8) [10] near ring N is alled a distributive near ring i a b ) ab a or all a, b, N. Theorem1.9)[11] Let N be a P-regular near ring.then or eah n N there exists n N suh that n n P. Theorem1.10) [11] Let N be a P-regular distributive near ring.then or every let ideal L and right ideal R o N. P R) P L) P RL. Deinition 1.11)[] Let N be a non- empty set a mapping : N [0,1 ] is alled a uzzy subset o N, where [0,1] is a losed interval o real numbers. Deinition 1.1) [4] Let be a non- empty uzzy subset o a near ring N,that is y) 0 or some y N ) then is said to be uzzy ideal o N i it satisies the ollowing onditions : 1) z y ) min{ z ), y )}; ) z. y ) min{ z ), y )} ; 3) y z y ) z ); 4) z. y ) y ), y,z N. When the subset o N satisies 1, is alled uzzy sub near ring. Deinition 1.13) [3] n intuitionisti uzzy set in a non-empty set X is an objet having the orm = { x, x), x)) x X}, where the untions : X [0,1 ] and : X [0,1 ] denoted the degree o membership and the degree o non membership o eah element respetively,and 0 x) x) or all x X. x X to the set, 1807

3 Deinition 1.14)[3] n intuitionisti uzzy set, ) in N is an intuitionisti uzzy subnear ring o N i or x, y N 1) x - y) min { x), y)} ) xyz) min { x), z)} 3) x - y) max{ x), y)} 4) xyz) max{ x), z)} Proposition 1.15) [1] Let X be a non- empty set. mapping : N [0,1 ] is a uzzy set in X, the omplement o,denoted by, is the uzzy set in X given by x) 1 x) or all x N or any I X, denote the harateristi untion o I. X I For any uzzy set μ and h [0,1], we deine two sets,, h) { x X x) h} and L, h) { x X x) h} Whih are alled upper and lower h-level ut o μ respetively, and an be used to the haraterization o μ. - bi-ideal with respet to an element o a near ring N In this setion we devoted to study bi-ideal with respet to an element r o a near ring N and give some properties, theorem about this ideal. Deinition.1) nonempty subset B o a near ring N is alled a bi-ideal with respet to an element o N and denoted by r-b-ideal o a near ring i 1) B together with addition is a subgroup o N. ) r. BNB r. B, r N. Example.). Let N= {0,a,b,} be the near ring deined by aleys + 0 b. 0 a b 0 0 b a 0 b 0 b 0 b b B C 0 a B C B a 0 C 0 b 0 b Then B={0,a} is b-bi-ideal sine b. BNB b. B Remark.3) I B 1 and B be two r-bi-ideal o near ring N, then B 1. B o N may be not r-bi-ideal Example.4) Considers the near ring N={ 0,a,b,} with addition and multipliation deined by the ollowing tables Let B 1 ={0,3}and B ={1,,3} are two 3-bi-ideal o N but B {3} is not two 3-bi-ideal o N. Remark.5) Not all r-bi-ideal o a near ring are bi-ideal o N. Example.6) B1 Consider the near ring N in example.) let B={0,b} be b-bi-ideal but B is not bi-ideal o N sine BNB B 1808

4 Theorem.7) Let N 1, +,.) and N,,. ) be two near rings, : N1 N be an epimomorphism and B be r- bi- ideal o N 1.Then B ) is r)- bi-ideal o N. Let B be r-bi- ideal o N 1, B) is a subgroup o N to prove B) is an r-bi-ideal o N Let r N1, suh that r) N,hene r. BNB rb Sine B is r- bi- ideal o N 1 r. BN1B) r. B) be an epimomorphism r). B). N1). B) r). B). r). B). N. B) r). B). B) is r)- bi- ideal o N Theorem.8) Let N 1, +,.) and N,,. ) be two near rings, and : N N be epiomomorphism and J be a 1 r)- bi-ideal o N. Then -1 J) is a r-bi- ideal o N 1, where y= r), ker 1 J). Let r N1, -1 J) is a subgroup o N 1 y. JN J y r. y. JN J) y). J ). J y N ). J) J ) y). J ) J ) N1 J ) r. J ) -1 J) is a r-bi-ideal o N 1. Remark.9) Not all r-bi-ideals o the near ring N are quasi ideal. Example.10). Let N= {0,a,b,} be the near ring deined by aleys + 0 b. 0 a b 0 0 b a 0 b a 0 b 0 b b B C 0 a b C B a 0 0 b 0 b Let B= {0, a} is -bi-ideal sine. BNB. B but is not quasi ideal Theorem.11) Let N be a P-regular near ring and B a r-bi-ideal o N. Then every y B There exist p P and b B suh that y p b. Let N be a P-regular near ring and B a r-bi-ideal o N and y B N, there exists z N suh that r yzy) y p or some p P thus y p ryzy) sine B is r-bi- ideal o N,we have r yzy) rbnb rb sine p P together with addition is a subgroup o N, we have p P put p p and b ryzy) thus y p r yzy) p b P B. Theorem.1) Let N be a P-regular distributive near ring and let B 1, B are a r-bi-ideals o N. i b B 1B and y N, then the element b an be represented as b p rb1 b and b1 b yp P or some p P, r, yi N, b1 B1 and b B. 1809

5 Let b B 1 B sine N is a P-regular near ring there exists y 1 N suh that rb b b P sine b B 1 B by theorem 1.9) we have b p 1 b1 or some p 1 P and b 1 B, b p b or some p P and b B,sine rb b b P,we have rb b b p3 or some p 3 P.Thus b p 3 rbb.hene = p r p b ) y p ) b p3 rp1 p rp1 b rb1 p rb1 b p3, rp1 p, rp1 b, rb1 p, rb1 b P sine P is an ideal o N,t Then p xp 3 1 p xp1 b xb1 p p4 p4 P Thus b p4 rb1 b rb1 b b p hene 1 1 yp b p4) yp byp p4 yp For some. So 4 rb y b P P P. 3-Intuitionisti uzzy bi- ideal with respet to an element o a near ring In this setion we devoted to study uzzy bi-ideal with respet to an element o a near ring N, we introdue the notion intuitionist uzzy bi-ideal with respet to an element o the near ring N, and give some properties, theorem about this ideals. Deinition 3.1) uzzy set o a near ring is alled uzzy bi- ideal with respet to an element o a near ring N i 1) rx - y)) min { rx), ry)} x, y N,r N. ) rxyz)) min { rx), rz)} x, y,z N,r N. It denoted by F-r-bi-ideal o N. Example 3.) Consider the near ring N= {0,1,,3} With addition and multipliation deined by the ollowing tables The uzzy subset o N whih is deined by i y 0,1 y) 0 otherwise is F--bi-ideal o N Deinition 3.3) n intuitionisti uzzy set, ) in N is an intuitionisti uzzy bi-ideal with respet to an element r o N i or all x, y, z N, r N, 1810

6 1) rx - y)) min { rx), ry)} ) rxyz)) min { rx), rz)} 3) rx - y)) max{ rx), ry)} 4) rxyz)) max{ rx), rz)} Theorem 3.4) n intuitionisti uzzy set, ) in N is an intuitionisti F-r- bi-ideal o N i and only i the uzzy set and are F-r- bi-ideal o N. I, ) is an intuitionisti F-r-bi-ideal o N.then learly is a F-r-bi-ideal o N, or all x, y N, rx - y)) 1 rx - y)) rxyz)) 1 max{ rx), ry)} min {1 rx),1 ry)} min{ rx), ry)} rxyz)) 1 max { rx), rz)} min {1 rx),1 rz)} min{ rx), rz)}, x, y N,r N, thus is a F-r- bi-ideal o N. onversely that and are F-r- bi-ideal o N, then learly the onditions 1, o deinition 3.3) are valid.now or all x, y N, r N. 1 rx - y)) rx - y)) min{ rx), ry)} 1- max { rx), ry)} ther ore rx - y)) max { rx), ry)} x, y,z N,r N, 1 rxyz)) rxyz)) min{ rx), rz)} therore max { rx), rz)} rxyz)) max { rx), rz)} Thus, ) is an intuitionisti F-r- bi-ideal o N. Theorem 3.5) n intuitionisti uzzy set, ) in N, is an intuitionisti F-r- bi-ideal o N i and only i, ) and, ) are intuitionisti F-r- bi-ideal o N. I, ) is an intuitionisti F-r- bi-ideal o N, then ) and are F-r-bi-ideal o N, rom theorem 3.4).Thereore, ) and, ) are intuitionisti F-r- bi-ideal o N. 1811

7 Conversely i, ),, ) are intuitionisti F-r- bi-ideal o N, then the uzzy sets and are F-r-bi-ideal o N,thereore, ) is intuitionisti F-r- bi-ideal o N. Proposition 3.6) n intuitionisti uzzy set, ) in N is a F-r-bi-ideal o N i and only i all non empty set, h) and, t) are r-bi-ideal o N or all o N h Im ) and t Im ) respetively. Suppose that, ) is an intuitionisti F-r-bi-ideal o N, or x, y, h),we have rx - y)) min{ rx), ry)} h Thereore r x y), h) let x, z, h) and y N. then rxyz)) min{ rx), rz)} h and so rxyz),h) hene, h) is a r- bi-ideal o N. For all h Im ).similarly we an show that, t) is also a r- bi-ideal o N or all t Im ).Conversely suppose that, h) and, t) are r- bi-ideal o N or all him ) and t Im ) respetively.suppose that x, y N and μ rx-y)) min{μ rx),μ ry)} Choose h suh that rx - y)) h min{ rx), ry)} Then we get x, y, h) r x y), h) a ontradition. Hene rx - y)) min{ rx), ry)}. similar argument shows that rxyz)) min{ rx), rz)} For all x, y, z N. likewise we an show that rx - y)) max{ rx), ry)} rxyz)) max{ rx), rz)} Hene, ) is an intuitionisti F-r- bi-ideal o N. Theorem 3.7) non empty set B o N is r- bi-ideal o N i and only i, ) is an intuitionisti F-r- biideal o N. Straight orward. Reerenes 1. Pilz, G Near rings, North Hollanda Dub. Co. New York.. Zadeh L Fuzzy Set. Inormation and ontrol., 8, pp : tanassov, K.T Intuitionisti uzzy sets, Fuzzy sets and system, 01), pp: Biswas, D.T.K Fuzzy Ideal O Near Ring. Bull. Cal.Math. Si., 89, pp: bujabal, H..S. 000.On Struture nd Commutativity O Near Rings, ntoagasta-chile, 19), pp: Vasantha, W.B. 00. Samarandahe near ring, United States o meria, pp: Davender, M.S Fuzzy homorphism o near rings. Fuzzy Set Systems, 45, pp: Yakabe, I Quasi-ideals in near rings. Mathematial reports College o General Eduation Kyushu University. 14, pp: phisit Muangma. 01. P-regular near ring haraterized by their bi ideals. University o phayao, Thailand, pp: Chelvam,T.T. and Ganesan,N On bi-ideals o near-rings. Indian Journal o Pure and pplied Mathematis, 18, pp: Choi, S.J P-regularity o a near ring, M.S. Thesis, university o Dong. B B but 181