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College Algebra Exam Review 283

College Algebra Exam Review 283 - domain if 1 2 R(b The...

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6.4. INTEGRAL DOMAINS 293 (c) Show that if S is an integral domain and ' W R ! S is a homo- morphism, then N ker .'/ . 6.4.2. If x is a nilpotent element in a ring with identity, show that 1 x is invertible. Hint: Think of the power series expansion for 1 1 t , when t is a real variable. 6.4.3. An element e in a ring is called an idempotent if e 2 D e . Show that the only idempotents in an integral domain are 1 and 0 . (We say that an idempotent is nontrivial if it is different from 0 or 1 . So the result is that an integral domain has no nontrivial idempotents.) 6.4.4. Show that if R is an integral domain, then the ring of polynomials ROExŁ with coefficients in R is an integral domain. 6.4.5. Generalize the results of the previous problem to rings of polyno- mials in several variables. 6.4.6. Show that if R is an integral domain, then the ring of formal power series ROEOExŁŁ with coefficients in R is an integral domain. What are the units in ROEOExŁŁ ? 6.4.7. (a) Show that a subring R 0 of an integral domain R is an integral
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Unformatted text preview: domain, if 1 2 R . (b) The Gaussian integers are the complex numbers whose real and imaginary parts are integers. Show that the set of Gaussian inte-gers is an integral domain. (c) Show that the ring of symmetric polynomials in n variables is an integral domain. 6.4.8. Show that a quotient of an integral domain need not be an integral domain. 6.4.9. Prove Lemma 6.4.3 . 6.4.10. Show that addition and multiplication on Q.R/ is well defined. This amounts to the following: Suppose a=b ³ a =b and c=d ³ c =d and show that .ad C bc/=bd ³ .a d C c b /=b d and ac=bd ³ a c =b d . 6.4.11. (a) Show that Q.R/ is a ring with identity element Œ1=1Ł and D Œ0=1Ł . (b) Show that Œa=bŁ D if, and only if, a D . (c) Show that if Œa=bŁ ¤ , then Œb=aŁ is the multiplicative inverse of Œa=bŁ . Thus Q.R/ is a field....
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