Talk:Localization (commutative algebra)

Dyadic and etymology

It seems that integers are being embedded into dyadic fractions, contrary to what is stated in the article.

of course you're right on the dyadic fractions. Concerning etymology, does the word come from turning rings into local rings? - 80.143.125.195 14:57, 18 January 2007 (UTC)[reply]

Total ring of fractions

I think the author has the wrong definition of "total ring of fractions," which I believe is a very specific localization, namely the localization of a ring with respect to the multiplicatively closed set of all non-zero-divisors in that ring. --66.92.4.19 (talk) 15:23, 23 October 2008 (UTC)[reply]

Inverting an ideal

A recent edit suggested inverting a multiplicative system containing an ideal. However, every ideal contains 0, so the localization at any multiplicative system containing an ideal is the zero ring. I think they just meant the semigroup containing a specific element, which might as well be written {f^n:n=0,1,...}. JackSchmidt (talk) 07:07, 31 December 2008 (UTC)[reply]

Microlocalization

I believe that the end of the article contains an error. Micro local analysis has nothing to do with (micro) localization, as far as I understand.

This has now been corrected.

Citation correction

I have a strange feeling the citation at the bottom should be Lang's Algebraic Number Theory. I know of no book by him entitled "Analytic Number Theory," and furthermore, the information in this article falls under algebraic number theory, not analytic. —Preceding unsigned comment added by 141.154.116.219 (talk) 23:23, 7 September 2008 (UTC)[reply]

Thank you for pointing this out: this has now been corrected.

Wrong property

First of the properties listed ($S^{-1}R = 0$ iff $0 \in S$) appears to be wrong. S may also contain nilpotent element. — Preceding unsigned comment added by 188.123.231.34 (talk) 23:20, 29 September 2011 (UTC)[reply]

No, the property is correct. Note that S contains a nilpotent element if and only if

0\in S

For example, if

s\in S

is nilpotent, then there exists a positive integer n such that sⁿ = 0 and since S is multiplicatively closed, it follows that

0\in S

.--PS T 00:27, 2 October 2011 (UTC)[reply]

Maximal ideal when localizing versus (the complement of ) a prime ideal

The example section says that when R is a commutative ring, and p is a prime ideal, then localizing against R-p yields a local ring R_p with maximal ideal p. But isn't the ideal really pR_p? Even when localization is 1-1, p is not in general an ideal of R_p under the inclusion.

Thomaso (talk) 19:26, 14 January 2013 (UTC)[reply]

You are right. I have corrected the sentence. D.Lazard (talk) 20:24, 14 January 2013 (UTC)[reply]

Use of the term "annihilator"

The use of the term "annihilator" for the ideal {a ∈ R; ∃s ∈ S : as = 0} in the section "For general commutative rings" is wrong. Any element of Ann(S) annihilates all of S (compare with Annihilator (ring theory)), but that's not what is needed in this situation. 2A02:810D:980:1704:4446:96B8:E70F:85A8 (talk) 08:43, 17 October 2014 (UTC)[reply]

~~There isn't technically anything wrong with the use of the word annihilator here, but~~ the explanation is a little muddled. I'll take a look at it tonight. Rschwieb (talk) 20:14, 17 October 2014 (UTC)[reply]

Also btw, {a ∈ R; ∃s ∈ S : as = 0} is rarely an ideal, ~~and I don't believe the text implies this at all.~~ The problem seems to be a disconnect between this statement and the use of Ann(S) in the rest of the paragraph. Rschwieb (talk) 00:11, 19 October 2014 (UTC)[reply]

Oh wait, upon rereading it I see of course the author did make that mistaken claim about the annihilator. I went ahead and took it out. Rschwieb (talk) 00:18, 19 October 2014 (UTC)[reply]

You confused ∃ with ∀. The annihilator is {a ∈ R; ∀s ∈ S : as = 0}. GeoffreyT2000 (talk) 18:01, 9 June 2015 (UTC)[reply]

Title of article

I propose the article be renamed 'Localisation (commutative algebra)' or similar Joel Brennan (talk) 16:14, 24 May 2019 (UTC)[reply]

I agree. I'll be bold and doing it. D.Lazard (talk) 17:19, 24 May 2019 (UTC)[reply]

Additions to the 'Properties' section

In the 'Properties' section (1.3), before the bullet point about the bijection between prime ideals in the ring and in the localisation, there should be a bullet point about the bijection between ordinary ideals of the ring and the localisation; the bijection of the prime ideals is then a restriction of this bijection. There should also be a bullet point afterwards saying that the bijection between ideals does not restrict to a bijection of maximal ideals. The reason I have not made this edit myself is because I do not know a counterexample for the maximal ideals. Joel Brennan (talk) 21:15, 29 May 2019 (UTC)[reply]

Two points.

Are you sure there is such a bijection for ideals not just prime ideals? If I recall, there is such one for primary ideals but not sure about ordinary ideals. In any case, one need a ref for such a statement.
I agree it’s a very good idea to mention the bijection doesn’t restrict to maximal ideals (counterexample I can think: R = 2-dimensional ring, finitely generated as an algebra over a field, and p height-one prime, then pR_p is maximal while p isn’t.) —- Taku (talk)

Let

I=p\cap q

the non-prime intersection of two ideals. Then the localization at

p

gives

I_{p}=p_{p}.

Similarly, if

t\in q

and

t\notin p,

one has also

I_{t}=p_{t}.

So there is no injectivity. The true result is that there is a bijection between the ideals of

S^{-1}R

and the ideals of

R

whose all associated prime ideals have an empty intersection with

S

. This is an immediate consequence of the following result, which should be added to the article:

The primary components of

S^{-1}I

are the localization at

S

of the primary components of

I

whose associated prime ideal do not intersect

S

.

Also, Taku's example may be generalized as: If p is any non-maximal prime ideal, then pR_p is maximal. Another example is: if R is a local ring of dimension higher that one, and S contains a non-unit non nilpotent element, then

S^{-1}R

has maximal ideals of the form

S^{-1}p

with

p

prime and non-maximal in

R

. D.Lazard (talk) 09:10, 30 May 2019 (UTC)[reply]

Isomorphic localization iff same saturation ?

It is stated in the article that : "If $S$ and $T$ are two multiplicative sets, then $S^{-1}R$ and $T^{-1}R$ are isomorphic if and only if they have the same saturation, or, equivalently, if $s$ belongs to one of the multiplicative set, then there exists $t\in R$ such that $st$ belongs to the other."

This appears to be false : take $S=\{0,1\}$ and $T=R$ , then the localizations are isomorphic (to the zero ring) but the saturation of $S$ is merely the set of zero divisors. Another example is $R=K\times K$ , and letting $S$ and $T$ be the complements of the maximal ideals $K\times 0$ and $0\times K$ . Then the localizations are isomorphic (to $K$ ) but $S$ and $T$ are already saturated and different from each other.

Did I miss something ? I think the usual theorem states that this is supposed to be an isomorphism of $R$ -algebras. EHecky (talk) 17:20, 20 March 2023 (UTC)[reply]

By definition of the saturation, if

0\in S,

then the saturation of

S

is

R

, since

x\cdot 0=0

for every

x\in R.

In other words,

x

divides 0, even when it is not a zero divisor. For zero divisors, the second factor must be nonzero. D.Lazard (talk) 17:49, 20 March 2023 (UTC)[reply]

Filled in Zariski open sets section

I'm not an algebraic geometer, so I offer somebody who is one to take a look. Svennik (talk) 14:24, 17 April 2024 (UTC)[reply]