9
$\begingroup$

I can give examples of non-noetherian rings having irreducible ideals that are not primary. Among them there are idealizations and valuation domains. But the first non-noetherian ring we are thinking about is $K[X_1,\dots,X_n,\dots]$, $K$ a field. The finitely generated ideals of this ring have primary decomposition, so if they are irreducible then are necessarily primary.

My question is the following:

Are there irreducible ideals that are not primary in $K[X_1,\dots,X_n,\dots]$?

$\endgroup$
2
+50
$\begingroup$

Another example using idealization construction (i.e. $(a,b)(c,d)=(ac,ad+bc)$):

  1. start with $R=k[T]_{(T)}+k(T)$, with $k=$prime field contained in $K$;
  2. primary ideals of $R$ are $(0)+(0)$, $(0)+k(T)$, and $(T^n)+k(T)$ $(n>0)$;
  3. the ideal $I=(0)+k[T]_{(T)}$ is irreducible and not primary;
  4. $R$ is countable so it's a quotient of $k[X_1,X_2,...]$;
  5. extend scalars from $k[X_1,X_2,...]$ to $K[X_1,X_2,...]$
$\endgroup$
  • $\begingroup$ This looks fine. The only step in doubt is (5) since it's not so clear (to me) that (non)primary, respectively irreducible ideals behaves well when extending in faithfully flat extensions. $\endgroup$ – user26857 Nov 13 '14 at 22:42
  • $\begingroup$ They behave well in this example. Extension of scalars gives a 1-1 correspondence between the ideal lattices i.e. $\endgroup$ – David Lampert Nov 14 '14 at 0:21
  • $\begingroup$ $(0)+(0),...,(0)+(T^3),(0)+(T^2),(0)+(T),(0)+(1),\ldots,(T^3)+(1),(T^2)+(1),(T)+(1),(1)+(1)$. (Sorry, I'm not fluent with math markup here.) $\endgroup$ – David Lampert Nov 14 '14 at 0:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.