Rings: Applications of the integers
In this chapter we assemble some results on rings which we obtain by using a specific knowledge of the natural numbers and the integers. We begin the chapter with some work refining our knowledge of finite and infinite sets. We then routinely study some theorems extending the associative, commutative, and distributive laws to any finite number of elements of a ring. We then extend to the integers the division algorithm earlier established for the natural numbers and discuss briefly prime numbers. After this we study the use in rings of the integers to indicate repeated additions and repeated multiplications: multiples and exponents. We consider in Section 4.5 the important result that every integral domain is included in some field. We show the existence of such a field and call it the field of fractions of the given integral domain. We specifically apply the theorem to the integers to construct the ordinary fractions or rational numbers. We finally, in Section 4.6, study the characteristic of a ring.
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