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v43-f10integersfinal

v43-f10integersfinal - Notes c F.P Greenleaf 2000-2010...

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Notes: c circlecopyrt F.P. Greenleaf, 2000-2010 v43-f10integers.tex (version 9/29/10) Algebra I Section 2: The System of Integers 2.1 Axiomatic definition of Integers. The first algebraic system we encounter is the integers. In this note we list the axioms that determine the system of integers, along with many simple consequences of those axioms. Most of those consequences will be stated without proof, or left as exercises; our main purpose in this section is to survey the facts about the integers you can safely assume in later discussions. Besides, the missing proofs will be handled later on in a more general context ( the theory of rings ). The integers are a system ( Z , + , · ) which consists of a set Z equipped with two operations (+) and ( · ) that map Z × Z Z . We now identify the properties these operations must possess to become the familiar system of integers. At first we shall consider a more general set R equipped with two operations (+) and ( · ) from R × R R . We will impose axiomatic conditions in stages, finally arriving at the axioms characteristic of the system of integers. 2.1.1 Axioms I: Commutative Ring. The system ( R, + , · ) is a commutative ring with identity if the operations have the following properties. A.1. ( x + y ) + z = x + ( y + z ) (associativity of addition) A.2. x + y = y + x (commutativity of addition) A.3. There exists an element 0 R such that 0 + x = x = x + 0 for all x R . This is the “zero element” of the system. A.4. Every element x R has an “additive inverse,” denoted by x , which has the property x + ( x ) = 0 = ( x ) + x . Later on we will see that this set of axioms, which govern the (+) operation only, makes ( R, +) into what algebraists would call a commutative group . We now add some axioms concerning multiplication and its interaction with addition. M.1. ( x · y ) · z = x · ( y · z ) (associativity of multiplication) M.2. x · y = y · x (commutativity of multiplication) M.3. There exists in R a “multiplicative identity element,” denoted by 1 (or some- times 1 R ), which has the characteristic property that 1 · x = x = x · 1 for all x R . M.4. x · ( y + z ) = x · y + x · z (distributive law) M.4’ ( x + y ) · z = x · z + y · z (distributive law) M.5. 1 negationslash = 0 (we exclude the “trivial ring,” which has 0 as its only element) square Below we list important consequences of this set of axioms. You will recognize many familiar attributes of the integers, but be aware that every one of these ancillary properties must be proved from the fundamental properties listed in Axioms I . This is not always a simple matter. It is suggested that you try making your own proofs for the items marked with ( ) in the following list. 2.1.2 Consequences of Axioms I. For a commutative ring with identity ( R, + , · ) we have 1. The xero element 0 is unique; if 0 is any other element in R such that x + 0 = x = 0 + x for all x R , then 0 = 0.
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