Are unique prime ideal factorization domains locally noetherian?
$begingroup$
In this question I asked: "Are unique prime ideal factorization domains noetherian?".
In this answer Badam Baplan pointed out that locally noetherian domains are unique prime ideal factorization domains (UPIFD), that certain non-noetherian domains are locally noetherian (the first example seems to have been given by N. Nakano in 1953), and thus that these non-noetherian domains are UPIFD.
This prompts the present question:
Are unique prime ideal factorization domains locally noetherian?
Edit. For the sake of completeness recall that a UPIFD is a domain satisfying the following condition:
If $mathfrak p_1,dots,mathfrak p_k$ are distinct nonzero prime ideals of $A$, and if $m$ and $n$ are distinct elements of $mathbb N^k$, then we have
$$
mathfrak p_1^{m_1}cdotsmathfrak p_k^{m_k}nemathfrak p_1^{n_1}cdotsmathfrak p_k^{n_k}.
$$
commutative-algebra maximal-and-prime-ideals noetherian integral-domain
asked Jan 5 at 11:47
Pierre-Yves GaillardPierre-Yves Gaillard
13.4k23184
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add a comment |
$begingroup$
In this question I asked: "Are unique prime ideal factorization domains noetherian?".
In this answer Badam Baplan pointed out that locally noetherian domains are unique prime ideal factorization domains (UPIFD), that certain non-noetherian domains are locally noetherian (the first example seems to have been given by N. Nakano in 1953), and thus that these non-noetherian domains are UPIFD.
This prompts the present question:
Are unique prime ideal factorization domains locally noetherian?
Edit. For the sake of completeness recall that a UPIFD is a domain satisfying the following condition:
If $mathfrak p_1,dots,mathfrak p_k$ are distinct nonzero prime ideals of $A$, and if $m$ and $n$ are distinct elements of $mathbb N^k$, then we have
$$
mathfrak p_1^{m_1}cdotsmathfrak p_k^{m_k}nemathfrak p_1^{n_1}cdotsmathfrak p_k^{n_k}.
$$
commutative-algebra maximal-and-prime-ideals noetherian integral-domain
asked Jan 5 at 11:47
Pierre-Yves GaillardPierre-Yves Gaillard
13.4k23184
$endgroup$
$begingroup$
In my opinion this property is not enough to imply noetherianity, not even locally.
$endgroup$
– user26857
Jan 5 at 19:32
$begingroup$
Crossposted on MathOverflow: mathoverflow.net/q/320787/461
$endgroup$
– Pierre-Yves Gaillard
Jan 13 at 12:48
$begingroup$
The question has been answered on MathOverflow by Dario Spirito: mathoverflow.net/a/321090/461
$endgroup$
– Pierre-Yves Gaillard
Jan 17 at 11:54
add a comment |
$begingroup$
In this question I asked: "Are unique prime ideal factorization domains noetherian?".
In this answer Badam Baplan pointed out that locally noetherian domains are unique prime ideal factorization domains (UPIFD), that certain non-noetherian domains are locally noetherian (the first example seems to have been given by N. Nakano in 1953), and thus that these non-noetherian domains are UPIFD.
This prompts the present question:
Are unique prime ideal factorization domains locally noetherian?
Edit. For the sake of completeness recall that a UPIFD is a domain satisfying the following condition:
If $mathfrak p_1,dots,mathfrak p_k$ are distinct nonzero prime ideals of $A$, and if $m$ and $n$ are distinct elements of $mathbb N^k$, then we have
$$
mathfrak p_1^{m_1}cdotsmathfrak p_k^{m_k}nemathfrak p_1^{n_1}cdotsmathfrak p_k^{n_k}.
$$
commutative-algebra maximal-and-prime-ideals noetherian integral-domain
asked Jan 5 at 11:47
Pierre-Yves GaillardPierre-Yves Gaillard
13.4k23184
$endgroup$
In this question I asked: "Are unique prime ideal factorization domains noetherian?".
In this answer Badam Baplan pointed out that locally noetherian domains are unique prime ideal factorization domains (UPIFD), that certain non-noetherian domains are locally noetherian (the first example seems to have been given by N. Nakano in 1953), and thus that these non-noetherian domains are UPIFD.
This prompts the present question:
Are unique prime ideal factorization domains locally noetherian?
Edit. For the sake of completeness recall that a UPIFD is a domain satisfying the following condition:
If $mathfrak p_1,dots,mathfrak p_k$ are distinct nonzero prime ideals of $A$, and if $m$ and $n$ are distinct elements of $mathbb N^k$, then we have
$$
mathfrak p_1^{m_1}cdotsmathfrak p_k^{m_k}nemathfrak p_1^{n_1}cdotsmathfrak p_k^{n_k}.
$$
commutative-algebra maximal-and-prime-ideals noetherian integral-domain
commutative-algebra maximal-and-prime-ideals noetherian integral-domain
asked Jan 5 at 11:47
Pierre-Yves GaillardPierre-Yves Gaillard
13.4k23184
asked Jan 5 at 11:47
Pierre-Yves GaillardPierre-Yves Gaillard
13.4k23184
edited Jan 5 at 12:33
Pierre-Yves Gaillard
asked Jan 5 at 11:47
Pierre-Yves GaillardPierre-Yves Gaillard
13.4k23184
asked Jan 5 at 11:47
Pierre-Yves GaillardPierre-Yves Gaillard
13.4k23184
asked Jan 5 at 11:47
Pierre-Yves GaillardPierre-Yves Gaillard
13.4k23184
13.4k23184
$begingroup$
In my opinion this property is not enough to imply noetherianity, not even locally.
$endgroup$
– user26857
Jan 5 at 19:32
$begingroup$
Crossposted on MathOverflow: mathoverflow.net/q/320787/461
$endgroup$
– Pierre-Yves Gaillard
Jan 13 at 12:48
$begingroup$
The question has been answered on MathOverflow by Dario Spirito: mathoverflow.net/a/321090/461
$endgroup$
– Pierre-Yves Gaillard
Jan 17 at 11:54
add a comment |
$begingroup$
In my opinion this property is not enough to imply noetherianity, not even locally.
$endgroup$
– user26857
Jan 5 at 19:32
$begingroup$
Crossposted on MathOverflow: mathoverflow.net/q/320787/461
$endgroup$
– Pierre-Yves Gaillard
Jan 13 at 12:48
$begingroup$
The question has been answered on MathOverflow by Dario Spirito: mathoverflow.net/a/321090/461
$endgroup$
– Pierre-Yves Gaillard
Jan 17 at 11:54
$begingroup$
In my opinion this property is not enough to imply noetherianity, not even locally.
$endgroup$
– user26857
Jan 5 at 19:32
$begingroup$
In my opinion this property is not enough to imply noetherianity, not even locally.
$endgroup$
– user26857
Jan 5 at 19:32
$begingroup$
Crossposted on MathOverflow: mathoverflow.net/q/320787/461
$endgroup$
– Pierre-Yves Gaillard
Jan 13 at 12:48
$begingroup$
Crossposted on MathOverflow: mathoverflow.net/q/320787/461
$endgroup$
– Pierre-Yves Gaillard
Jan 13 at 12:48
$begingroup$
The question has been answered on MathOverflow by Dario Spirito: mathoverflow.net/a/321090/461
$endgroup$
– Pierre-Yves Gaillard
Jan 17 at 11:54
$begingroup$
The question has been answered on MathOverflow by Dario Spirito: mathoverflow.net/a/321090/461
$endgroup$
– Pierre-Yves Gaillard
Jan 17 at 11:54
add a comment |
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$begingroup$
In my opinion this property is not enough to imply noetherianity, not even locally.
$endgroup$
– user26857
Jan 5 at 19:32
$begingroup$
Crossposted on MathOverflow: mathoverflow.net/q/320787/461
$endgroup$
– Pierre-Yves Gaillard
Jan 13 at 12:48
$begingroup$
The question has been answered on MathOverflow by Dario Spirito: mathoverflow.net/a/321090/461
$endgroup$
– Pierre-Yves Gaillard
Jan 17 at 11:54