Re: Basic questions on prime ideals,.

In article <dlt0j0$2me3$1@xxxxxxxxxxxxxxxxx>, James <James545@xxxxxxxxx> wrote:
>
>"Arturo Magidin" <magidin@xxxxxxxxxxxxxxxxx> wrote in message
>news:dlt09q$jsp$1@xxxxxxxxxxxxxxxxxxxxx
>> In article <dlsvhd$2mfm$1@xxxxxxxxxxxxxxxxx>, James <James545@xxxxxxxxx>
>> wrote:

[.snip.]

>> Thus, the prime ideals of A/I are in one-to-one inclusion preserving
>> correspondence with the prime ideals of A that contain I. (i.e., the
>> canonical correspondence respects primality).
>>
>
>Then why did my commutative algebra professor say that V( I ) is in one to
>one correspondence with Spec(A / I) only if I = radical of I? Here, V( I )
>is the (closed) set of prime ideals in Spec(A) containing I. He said it
>wasn't true if I was just a general ideal.


Spec(A/I) is a topological space, whose elements are the prime ideals
of A/I (hence, the prime ideals of A that contain I). V(I) is the
closed subset of Spec(A) consisting of the prime ideals of A that
contain I. There is natural map Spec(A/I) --> Spec(A) induced by the
canonical projection A-->A/I. Since this map is surjective, the map
induced on Spec(A/I) --> Spec(A) will be a homeomorphism of Spec(A/I)
onto the closed subset V(I) of X. (In general, if f:A->B is a
surjection, then the induced map Spec(B)-->Spec(A) is a homeomorphism
of Spec(B) onto V(ker(f)).

So I don't know *why* your professor said that. It could be that he is
trying to say something other than what your or I are interpreting.

By the by, if you are working with commutative rings with 1, then the
original problem is even easier: remember that an ideal I is prime (in
this setting) if and only if A/I is an integral domain. If P is an
ideal of A that contains I, then A/P is isomorphic to (A/I)/(P/I), so
P is a prime ideal if and only if A/P is an integral domain if and
only if (A/I)/(P/I) is an integral domain if and only if P/I is a
prime ideal of A/I.



--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.

Relevant Pages

  • Re: Notes on irreducibles, reducibles in Integral Domains
    ... take an integral domain R with ... R has only two prime ideals. ... Localize A at a non-zero prime ideal; if the resulting ring R is not ... irreducibles are at least existing. ...
    (sci.math)
  • Direct sum of rings
    ... Let R be a local ring with more than one nonzero prime ideals (i.e. ... prime ideals that arent.. ... in case R were an integral domain, ...
    (sci.math)