Unformatted text preview: R but is not a prime ideal. 6. Section 7.4, Exercise 10: Assume R is commutative. Prove that if P is a prime ideal of R and P contains no zero divisors then R is an integral domain. 7. Section 7.4, Exercise 16: Let x 416 be an element of the polynomial ring E = Z [ x ] and use the bar notation to denote passage to the quotient ring Z [ x ] / ( x 416) . (a) Find a polynomial of degree ≤ 3 that is congruent to 7 x 1311 x 9 +5 x 52 x 3 +3 modulo ( x 416) . (b) Prove that x2 and x + 2 are zero divisors in E . 8. Let R be an integral domain with identity. Prove that if R has ﬁnitely many elements then R is a ﬁeld. 9. Section 7.5, Exercise 4: Prove that any subﬁeld of R must contain Q . 1...
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 Fall '08
 OGUS
 Math, Algebra, Ring, Integral domain, Ring theory, Commutative ring, Principal ideal domain

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