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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