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 Ap. 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 xi 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.
This page was last updated on 25 Oct 2001.