Comments for Homework 6

Comments:

• Scores for the problems are as follows (total score=14):
  • II.3: 5.
  • II.6: 5.
  • all other problems: 1 each.
  
  
• II.3 is very well-done, however I noticed that the vast majority of the answers forgot to prove that the ideal { a/s | a in P, s not in P } is not the whole ring A_p. I had wanted to deduct a point for this oversight, but decided that it's probably quite pointless.
  
  
• For II.6, the key point is that A_(p) has exactly one prime element (of course you don't have to state this explicitly in the answer). Most of the errors occur in the following categories:
  
  
  • Claim: if x_i generate the ideal I in a UFD (factorial ring), then the gcd of the elements generate the ideal I. Unfortunately, this statement is precisely equivalent to the claim that the ring is PID. For example, if we take A = k[x, y], and let I = (x, y), then I is not generated by gcd(x, y) = 1.
    
    
  • Claim: a subideal of a principal prime ideal in a UFD is principal. Again not true: if we take A as above, and I = (x), J = (x)(x, y), then J is not principal.
  
  
  

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This page was last updated on 25 Oct 2001.