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Boolean Like Semirings Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 13, 619 - 635 Boolean Like Semirings K. Venkateswarlu Department of Mathematics, P.O. Box. 1176 Addis Ababa University, Addis Ababa, Ethiopia [email protected] B. V. N. Murthy Department of Mathematics Simhadri Educational Society group of institutions Sabbavaram Mandal, Visakhapatnam, AP, India [email protected] N. Amarnath Department of Mathematics Praveenya Institute of Marine Engineering and Maritime Studies Modavalasa, Vizianagaram, AP, India [email protected] Abstract In this paper we introduce the concept of Boolean like semiring and study some of its properties. Also we prove that the set of all nilpotent elements of a Boolean like semiring forms an ideal. Further we prove that Nil radical is equal to Jacobson radical in a Boolean like semi ring with right unity. Mathematics Subject Classification: 16Y30,16Y60 Keywords: Boolean like semi rings, Boolean near rings, prime ideal , Jacobson radical, Nil radical 1 Introduction Many ring theoretic generalizations of Boolean rings namely , regular rings of von- neumann[13], p- rings of N.H,Macoy and D.Montgomery [6 ] ,Assoicate rings of I.Sussman k [9], p -rings of Mc-coy and A.L.Foster [1 ] , p1, p2 – rings and Boolean semi rings of N.V.Subrahmanyam[7] have come into light. Among these, Boolean like rings of A.L.Foster [1] arise naturally from general ring dulity considerations and preserve many of the formal 620 K. Venkateswarlu et al properties of Boolean ring. A Boolean like ring is a commutative ring with unity and is of characteristic 2 with ab (1+a) (1+b) = 0. It is clear that every Boolean ring is a Boolean like ring but not conversely (see example 1.9 ).Later Swaminathan [ 10,11,12 ] has demonstrated that every Boolean like ring is a direct product of Boolean ring and a module over a Boolean ring. Further he investigated the study on algebraic and geometry of Boolean like rings extensively. Subrahmanyam N.V.[ 7 ] introduced the concept of Boolean semi rings and studied many geometric properties. In fact Boolean semi rings are nothing but a special class of near rings. In this paper our objective is to introduce the concept of Boolean like semi ring which is a generalization to Boolean like rings of A.L. Foster[ 2] and also a special class of near rings. Further we study its properties and also furnish examples that the two systems Boolean semirings of Subrahmanyam and Boolean like semi rings of our system are independent. 1. PRELIMINARIES In this article we recall certain definitions and results concerning Boolean semi rings and Boolean like rings. Further we prove some motivating results to introduce the concept of Boolean like semi rings. Definition [7] A system ( R,+, . ) a Boolean semi ring if and only if the following properties hold : 1. (R,+) is an additive (abelian) group ( whose “zero” will be denoted by “0”.) 2. (R, .) is a semi group of idempotents in the sense, aa=a ,for all a R 3. a(b+c) = ab + ac and 4. abc = bac , for all a,b,c R. Example [7]. Let (G,+) be any abelian group and define ab = b , for all a,b G. Then (G,+,.) is a Boolean semi ring . Here we give some examples which motivates us to introduce the concept of Boolean like semi rings Example 1.1. Consider the near ring defined on the Klein’s four group with N= {0,a,b,c}. “ + ” and “. “ defined by Boolean like semirings 621 + 0 a b c . 0 a b c 0 0 a b c 0 0 0 0 0 a a 0 c b a 0 a b c b b c 0 a b 0 a b c c c b a 0 c 0 a b c Then (N,+,.) is a Boolean semi ring. Note that (a+b)c ≠ ac + bc, ab ≠ ba , Characteristic of N = 2 In the sense that N is not commutative and right distributive law fails Also ab(a+b+ab) = ab is not true for all a,b N , since ac(a+c+ac) = a ≠ ac Example 1.2 Let M = {0,1,2} , “+” denotes addition modulo 3 and “.” Defined as follows + 0 1 2 . 0 1 2 0 0 1 2 0 0 1 2 1 1 2 0 1 0 1 2 2 2 0 1 2 0 1 2 Then (M,+, . ) is a Boolean semi ring. Note that Chr M ≠ 2 and ab(a+b+ab) ≠ ab as 0.1(0+1+0.1) = 2 ≠ 0.1 Example 1.3 .Let R = {0,x,y,z } , + and . are defined by + 0 x y z . 0 x y z 0 0 x y z 0 0 0 0 0 x x 0 z y x 0 x 0 x y y z 0 x y 0 0 0 0 z z y x 0 z 0 z 0 z Then clearly (R,+,.) is a (Left) near ring in which Chr = 2 and 622 K. Venkateswarlu et al ab(a+b+ab) =ab ,for all a,b R. Clearly it is not a Boolean semi ring since y2 ≠ y Example 1.4. Let K = {0,1,2,3} , “+” denotes addition modulo 4 and “.” defined by + 0 1 2 3 . 0 1 2 3 0 0 1 2 3 0 0 0 0 0 1 1 2 3 0 1 0 1 2 3 2 2 3 0 1 2 0 1 2 3 3 3 0 1 2 3 0 0 0 0 Then (M,+, .) is a(left) near ring , but not Boolean semi ring and Chr M ≠ 2 and also ab(a+b+ab) = ab is not true for all a,b M , as 1.2 (1+2+1.2) =1 ≠ 1.2 Example 1.5. Let L = {0,1,2,3} , “+” denotes addition modulo 4 and “.” Defined by + 0 1 2 3 . 0 1 2 3 0 0 1 2 3 0 0 0 0 0 1 1 2 3 0 1 0 1 2 3 2 2 3 0 1 2 0 2 0 2 3 3 0 1 2 3 0 1 2 3 Clearly (L,+, .) is a (left) near ring in which ab(a+b+ab) = ab , for all a,b L and Chr L ≠ 2 and which is not a Boolean semi ring and also x2 = x is not true for all x L , as 2.2 =0 Definition [ 4 ]. A near ring R is called Boolean near ring if x2 = x ,for all x R Example [ 4 ]. Let (Z, +) be a group and define ab = b, for all a,b Z. Then (Z,+,.) is a Boolean Near ring. We have the following in the Theorem 1.6. If R is a Boolean Near ring with ab(a+b+ab) = ab for all a R then characteristic R is 2 Proof. Let a = b in ab(a+b+ab) =a. Then aa(a+a+aa) = aa .Since a2 = a, we have that a( a+a+a ) = a ⇒ aa + aa + aa = a ⇒ a+a+a = a ⇒ a+a = 0 We observe an important fact in the following Remark 1.7 . If R is a near ring with characteristic 2 in which ab(a+b+ab)=ab holds for all a,b in R but still R may not be a Boolean near ring. Boolean like semirings 623 For instance, look into the example 1.3 , it is clear that ab(a+b+ab)=ab holds for all a,b in R but y2 ≠ y Definition [2 ]. A Boolean like ring R is a commutative ring with unit element in which for all elements a,b in R, a+a =0, ab(1+a)(1+b)=0. Remark 1.8[2]. Clearly every Boolean ring with 1 is a Boolean like ring , but not conversely. Example [10] Let R = {0,1,p,q} , “+”and “ .” are defined by + 0 1 p q . 0 1 P q 0 0 1 P q 0 0 0 0 0 1 1 0 q p 1 0 1 p q p p q 0 1 p 0 p 0 p q q p 0 1 q 0 q p 1 Then (R,+,.) is a Boolean Like ring but not Boolean ring as o = p2 ≠ p. Motivated from the above examples with different properties we are in a position to introduce the concept of Boolean like semi rings. 2 Boolean like semi rings and its properties We introduce the concept of Boolean like semi ring and give some examples of a Boolean like semi ring .we obtain certain properties of Boolean like semi rings .Also we introduce the concept of ideal in a Boolean like semi ring and obtain the various properties of ideals. Further we prove that every Jacobson radical is a nil radical in any Boolean like semi ring. Now we begin with the following Definition 2.1 . A non empty set R together with two binary operations + and . satisfying the following conditions is called a Boolean like semi ring 1.( R,+ ) is an abelian group 2. (R , . ) is a semi group 3. a.( b+c) = a.b +a.c for all a, b, c ∈ R 4. a + a = 0 for all a in R 5. ab( a+b+ab ) =ab for all a, b ∈ R. 624 K. Venkateswarlu et al Remark 2.2 1. Condition 3 is called left distributive law and hence a system (R,+,.) satisfying Conditions 1 , 2 and 3 is called a (left) near ring 2. From condition 4 of the above definition , it is clear that the near ring R is of characteristic 2 3.
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