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homework9

# homework9 - R but is not a prime ideal 6 Section 7.4...

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Math 113 Section 5 Homework #9 Fall 2010 Due: Monday, November 8 1. Determine all the ideals of Z / 12 Z . Determine which ideals are prime and which are maximal. 2. (a) Describe the ring structure on Z × Z . Does this ring have identity? (b) Describe all ring homomorphisms of Z × Z Z . 3. Section 7.4, Exercise 7: Let R be a commutative ring with 1 . Prove that the principal ideal generated by x in the polynomial ring R [ x ] is a prime ideal if and only if R is an integral domain. Prove that ( x ) is a maximal ideal if and only if R is a field. 4. Section 7.4, Exercise 8: Let R be an integral domain. Prove that ( a ) = ( b ) for some elements a, b R , if and only if a = ub for some unit u of R . 5. Section 7.4, Exercise 9: Let R be the ring of all continuous functions on [0 , 1] and let I be the collection of functions f ( x ) with f ( 1 3 ) = f ( 1 2 ) = 0 . Prove that I is an ideal of
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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 4-16 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 4-16) . (a) Find a polynomial of degree ≤ 3 that is congruent to 7 x 13-11 x 9 +5 x 5-2 x 3 +3 modulo ( x 4-16) . (b) Prove that x-2 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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