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C-Simple and Bi-Ideal Simple Semirings The Islamic University of Gaza Deanery of Higher Studies Faculty of Science Department of Mathematics C-SIMPLE AND BI-IDEAL SIMPLE SEMIRINGS Presented By Saeed Nouh Saeed Al-Zayyan Supervised By Dr. Ahmed Al-Mabhouh SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE AT THE ISLAMIC UNIVERSITY OF GAZA GAZA, PALESTINE JUNE 2006 °c Copyright by Saeed Nouh Saeed Al-Zayyan, 2006 To My parents, wife, sons, and my sincere friends, and to all knowledge seekers. ii Table of Contents Table of Contents iii Abstract iv Acknowledgements v Introduction 1 1 Preliminary Notions 3 1.1 Groups, Rings, Semigroups, Semirings, and Relations . 3 1.2 Ideals and Bi-Ideals of a Semirings . 11 1.3 Congruence Relations . 13 2 Bi-Ideal-Simple Semirings 20 2.1 Bi-Ideal-Simple Semiring-Introduction . 20 2.2 Additively Zeropotent Semirings . 25 2.3 Bi-Ideal Simple Zeropotent Semirings . 30 3 Congruence-Simple Semirings 33 3.1 Congruence-Simple Commutative Semirings . 33 3.2 Congruence-Simple Additively Commutative Semirings of Finite Order 44 4 Smaradache Semirings 56 4.1 De¯nition of Smarandache Semirings and Examples . 56 4.2 Smarandache Special Elements in Semirings . 61 conclusions 66 References 67 iii Abstract In this research we survey and introduce some concepts of semirings, as a bi-ideal simple semirings, congruence simple semirings, and bi-ideal simple zeropotent semi- rings. Finally we introduce smarandache semirings, smarandache zero divisors, and we classify all the zero divisors in spacial semirings. iv Acknowledgements My special thanks after thanking Allah almighty, to the Department of Mathematics in the Islamic University of Gaza, specially Dr: Ahmed Al-Mabhouh for his great concern and support. Also, I thank my family for providing a comfortable environ- ment to me, to my sincere friends for their endless support. Finally, I would like to thank everyone who contributed in achieving this thesis. v Introduction The notion of a semiring (i,e., a universal algebra with two associative binary op- erations, where one of them distributes over the other) was introduced by Vandiver [6] in (1934). Needless to say, semiring found their full place in mathematics long before that year (e.g., the semirings of positive elements in ordered rings) and even more after (e.g., Various applications in theoretical computer science and algorithm theory). Congruence simple algebras (i.e., those possessing just two congruence relations) serve a basic construction material for any algebraic structure. The study of congru- ence simple commutative semiring with unity was introduced by mitchell and Fenoglio [11] in 1988. See also [12] In this thesis, we study congruence simple semirings and bi-ideal simple semirings. See [10], [13]. This thesis is divided into four chapters. In chapter1, we introduce some basic de¯nition, and results about ideals, bi-ideal and congruence relations of semirings. For more informations, see [1], [2], [8], [10], [11], [12], and [13]. Chapter2 is devoted for the study of bi-ideals simple semirings. See [13]. In Chapter3, Congruence-simple semirings are studied in more details. Also, we classify the congruence-simple commutative semirings. See [10], [1]. 1 2 In chapter4, we introduce the concept of Smarandache semirings. We give some de¯nitions and examples about this topic. See [14], [15], [16]. Chapter 1 Preliminary Notions In this chapter we introduce basic de¯nitions, theorems and results that would be useful for the rest of the thesis. More information can be found in [6], [8], [10], [11], [12], [15]. 1.1 Groups, Rings, Semigroups, Semirings, and Relations De¯nition 1.1.1 Let S be a non-empty set, we call S a group under the operation (¤), if the following properties hold 1. S is Closed under (¤), i:e for all a; b 2 S we have a ¤ b 2 S. 2. The operation (¤) is associative, i.e a ¤ (b ¤ c) = (a ¤ b) ¤ c for all a; b; c 2 S 3. There exists an auniqe element e in S such that e ¤ a = a ¤ e = a, for all a 2 S 3 4 we call it an identity for (¤). 4. For each a 2 S, there exists an element b 2 S such that a ¤ b = e = b ¤ a; b is called an inverse for a, and a is called an inverse for b. In addition, if a ¤ b = b ¤ a for all a; b 2 S, then S is called an abelian group. In the above de¯nition, if only conditions 1,2 are satis¯ed, then S is called a semigroup. (S; ¤) is a cancellative semigroup, if a ¤ b = a ¤ c, then b = c. De¯nition 1.1.2 Let S be a group, let H ⊆ S, we call H is a subgroup if H is a group under the same operation of S. De¯nition 1.1.3 Let (S; +;:) be a non-empty set with two binary operations, called addition and multiplication, and denoted by (+) and (.). Then S is called a ring with respect to these operations if the following properties hold. 1. (S; +) is an abelian group, 2. (S; :) is a semigroup, 3. For all a; b; c 2 S, we have a:(b + c) = a:b + a:c, and (a + b):c = a:c + b:c. A ring S has an identity element if there is an element in S, say 1 such that 1:x = x:1 = x for all x 2 S 5 Let S be a commutative ring with identity 1. An element a 2 S is said to be invertible if there exists an element b 2 S such that ab = 1. The element a is called a unit of S, and its multiplicative inverse is usually denoted by a¡1. De¯nition 1.2.4 Let S be a commutative ring with identity, and let a 2 S. The ideal < a >= Sa = fx 2 S : x = ra for some r 2 Sg is called the principal ideal generated by a. De¯nition 1.1.5 If S is commutative ring with identity 1, and each x 2 S has a multiplicative inverse, we call it a ¯eld. De¯nition 1.1.6 Let S be a non-empty set, with two operations (+, .), we call S a semiring if the following properties hold. 1. (S; +), (S; :) are semigroups. 2. 8a; b; c 2 S, we have a:(b + c) = a:b + a:c and (a + b):c = a:c + b:c. A semiring S need not have additive or multiplicative identity. A semigroup (S; ¤) is commutative i® 8a; b 2 S; a ¤ b = b ¤ a. We call a semiring S commutative if S(+), S(:) are both commutative, jSj will denote the cardinality of S. 6 An element ® of a semiring S is called additively absorbing if ® + x = x + ® = ® 8x 2 S, and multiplicatively absorbing if ®:x = x:® = ®8x 2 S. An element 1 of a semiring S is called an in¯nity or (bi-absorbing element)if it is both additively and multiplicatively absorbing. An element x 2 S is a multiplicative idempotent, if x:x = x2 = x, and an additive idempotent, if x + x = x. De¯nition 1.1.7 A non-trivial commutative semiring S is said to be a semi¯eld if there exists an element w 2 S and (T;:) is a group such that Sw == fwg and T = Snfwg. Theorem 1.1.8 Let (S; ¤) be a ¯nite cancellative commutative semigroup, then (S; ¤) is a group. Proof. Let jSj = n; S = fx1; x2; :::; xng, let a 2 S, and consider the set fa ¤ x1; a ¤ x2; :::; a ¤ xng. Note that if a ¤ xi = a ¤ xj, then xi = xj, since (S; ¤) is cancellative, then S = fa ¤ x1; a ¤ x2; :::; a¤ xng, since a 2 S, then 9xt 2 S; t · n, such that a ¤ xt = a (1.1.1) Now we will show the element xt is the identity, so let y 2 S, then y = a ¤ xj, so y ¤ xt = (a ¤ xj) ¤ xt = (a ¤ xt) ¤ xj = a ¤ xj = y, then xt is the identity element in (S; ¤), for any a 2 S, we have xt = a ¤ xj, so (S; ¤) closed under inverse. Note that an additive identity in a semiring need not be multiplicatively absorbing, see the example below 7 Example 1.1.9 Let S = [1; 1) ½ R+ (the set of positive real number), and (S; max; :), where (.) denotes the usual multiplication, a + b = max(a; b) , clearly that (S; max; :) is a semiring, note that 1 is the additive identity of S, since 1 + a = max(1; a) = a, for all 1 · a, a 2 R and not a multiplicatively absorbing, since 1:a = a, for all a 2 S If however a semiring has a multiplicatively absorbing additive identity we call it a zero and denote it by o, If has additive identity (additive zero) we call it 0. A semiring with additive zero is called zero-sum free if for all a; b 2 S, a + b = 0 implies a = b = 0: Note that o=0 in the set of real numbers, rational numbers, and integers. De¯nition 1.1.10 For a multiplication abelian group G, set V (G) = G[f1g. Extend the multiplication of G to V (G) by the rule x1 = 1x = 1 for all x 2 V (G). De¯ne an addition on V (G) by the rule x + x = x, x + y = 1 for all x; y 2 V (G) with x 6= y. Example 1.1.11 + Let Z± = Z [ f0g, note that (Z±; +;:) is a zero sum-free semiring of in¯nite cardi- nality. Example 1.1.12 Let S be a semiring, Mn£n = f(aij); aij 2 Sg be the set of all n £ n matrices.
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