College Algebra Exam Review 280

College Algebra Exam Review 280 - 290 6 RINGS Example...

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290 6. RINGS Example 6.4.2. (a) The ring of integers Z is an integral domain. (b) Any field is an integral domain. (c) If R is an integral domain, then RŒxŁ is an integral domain. In particular, KŒxŁ is an integral domain for any field K . (d) If R is an integral domain and R 0 is a subring containing the identity, then R 0 is an integral domain. (e) The ring of formal power series RŒŒxŁŁ with coefficients in an integral domain is an integral domain. There are two common constructions of fields from integral domains. One construction is treated in Corollary 6.3.14 and Exercise 6.3.7 . Namely, if R is a commutative ring with 1 and M is a maximal ideal, then the quotient ring R=M is a field. Another construction is that of the field of fractions 1 of an integral do- main. This construction is known to you from the formation of the rational numbers as fractions of integers. Given an integral domain R , we wish to construct a field from symbols of the form a=b , where a;b 2 R and b ¤ 0 . You know that in the rational numbers,
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