Microsoft Word - MODULE II More Number Theory.pdf

Microsoft Word - MODULE II More Number Theory.pdf - 1...

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1 MODULE II More Number Theory and Some Algebra ; n ( mod n) Ζ Ζ As we already know ( , +, ) has "algebraic" structure. Indeed. both + and are binary operations on that are associative and commutative; 0 is the identity of ( , +), 1 is the identity of ( , ) and Ζ Ζ Ζ Ζ ringoperator ringoperator ringoperator n the operations are linked by the distributive law a (b + c) = a b + a c. We put these ideas in a more general context and begin a discussion of , the integers module n, in this algebraic setting. Defi Ζ ringoperator ringoperator ringoperator nition 1 A binary operation on a non-empty set S is a function : S x S S. The operation * is said to be associative if a, b, c S (a b) c = a (b c)) If is associati * * * * * * * ve then (S, ) is called a semigroup. An element e is called an identity provided a e = e a = a a S. * * * ∀ ∈ 1 1 2 2 If an identity exists it is unique (indeed e = e e = e ) and (S, ) is referred to as a monoid. An element a is said to be invertible (or to have an inverse) if b such that a * * b = b a = e * { } -1 If an inverse exists it is unique (Proof - Exercise 1) and is denoted by a . If all elements of the monoid (S, ) are invertible then (S, ) is a group (Exercise 2: If T = a S a has an inverse then (T, ) * * * is a group). If is commutative, i.e. a, b S (a b = b a) * * * (S, ) is referred to as a commutative semigroup or commutative monoid or commutative group as the case may be. If the operation is denoted by + and (S, +) is a commutative group it is usually referre * d to as an abelian group. If the non-empty set R is equipped with two binary operations + and , the triple (R, +, ) is called a ring if (i) (R, +) is an abelian group with identity 0 (inv ringoperator ringoperator erses are denoted with minus signs, i.e. the inverse of a has the name -a) (ii) (R, ) is a semigroup and (iii) (distributive laws) a, b, c R a (b+ c) = a b + a c and ringoperator ringoperator ringoperator ringoperator (b + c) a = b a + c a ringoperator ringoperator ringoperator
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2 Exercise 3. (Submit this one) If (R, +, )is a ring then a R a 0 = 0 a = 0 Thus if (R, ) has an identity, say 1, and R 2 then 1 0 Prove these contentions. If R 2 and (R, ) is a ringoperator ringoperator ringoperator ringoperator ringoperator { } monoid, then (R, +, ) is a ring with identity. An element a R, where (R, +, ) is a ring with identity, is called a unit if it has a multiplicative inverse. The set of units U R - 0 and (U, ) is a g ringoperator ringoperator ringoperator roup. It is called the group of units of R. Exercise 4. Prove 0 U and (U, ) is group If (R, ) is a commutative semigroup then (R, +, ) is called a commutative ring. A commutative ring with identity su ringoperator ringoperator ringoperator { } ch that U = R - 0 is a field. { } Remark1. ( , +, ) is a commutative ring with identity such that U = 1, -1 . Ζ ringoperator + Definition 2. Let n . We write a b (mod n) if and only if n a-b Terminology: a is congruent to b modulo n Observation Congruence modulo n is an equivalence relation. Each equivalence class contain ∈ Ζ { } s a unique element from 0, 1,..., n-1 called the least residue of the class and is determined by the division algorithm, i.e. the least residue of the equivalence class containing a
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