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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
210:
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
244:
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.
709:
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.) —-
190:
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.
998:
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
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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.
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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.
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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.
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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.
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of course you're right on the dyadic fractions. Concerning etymology, does the word come from turning rings into local rings? -
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First of the properties listed ($ S^{-1}R = 0$ iff $ 0 \in S$ ) appears to be wrong. S may also contain nilpotent element.
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510:= 0} in the section "For general commutative rings" is wrong. Any element of Ann(S) annihilates all of S (compare with
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The problem seems to be a disconnect between this statement and the use of Ann(S) in the rest of the paragraph.
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It seems that integers are being embedded into dyadic fractions, contrary to what is stated in the article.
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of this bijection. There should also be a bullet point afterwards saying that the bijection between ideals
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Did I miss something ? I think the usual theorem states that this is supposed to be an isomorphism of
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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So there is no injectivity. The true result is that there is a bijection between the ideals of
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are isomorphic if and only if they have the same saturation, or, equivalently, if
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I propose the article be renamed 'Localisation (commutative algebra)' or similar
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I'm not an algebraic geometer, so I offer somebody who is one to take a look.
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Maximal ideal when localizing versus (the complement of ) a prime ideal
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the explanation is a little muddled. I'll take a look at it tonight.
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the non-prime intersection of two ideals. Then the localization at
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whose all associated prime ideals have an empty intersection with
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Thank you for pointing this out: this has now been corrected.
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belongs to one of the multiplicative set, then there exists
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is merely the set of zero divisors. Another example is
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are already saturated and different from each other.
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490:The use of the term "annihilator" for the ideal {
530:There isn't technically anything wrong with the
989:whose associated prime ideal do not intersect
470:You are right. I have corrected the sentence.
1093:Isomorphic localization iff same saturation ?
8:
1368:. Then the localizations are isomorphic (to
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364:is multiplicatively closed, it follows that
300:contains a nilpotent element if and only if
602:You confused ∃ with ∀. The annihilator is {
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679:Additions to the 'Properties' section
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663:I agree. I'll be bold and doing it.
106:This article is within the scope of
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412:The example section says that when
49:It is of interest to the following
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1605:Low-priority mathematics articles
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534:of the word annihilator here, but
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126:Knowledge:WikiProject Mathematics
1208:This appears to be false : take
129:Template:WikiProject Mathematics
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146:This article has been rated as
446:is not in general an ideal of
290:23:20, 29 September 2011 (UTC)
1:
985:of the primary components of
486:Use of the term "annihilator"
435:. But isn't the ideal really
261:23:23, 7 September 2008 (UTC)
236:This has now been corrected.
221:07:07, 31 December 2008 (UTC)
120:and see a list of open tasks.
1600:C-Class mathematics articles
897:{\displaystyle I_{t}=p_{t}.}
799:{\displaystyle I_{p}=p_{p}.}
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201:15:23, 23 October 2008 (UTC)
181:14:57, 18 January 2007 (UTC)
1301:{\displaystyle R=K\times K}
416:is a commutative ring, and
402:00:27, 2 October 2011 (UTC)
18:Talk:Localization of a ring
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1496:{\displaystyle x\cdot 0=0}
1397:17:20, 20 March 2023 (UTC)
948:The primary components of
854:{\displaystyle t\notin p,}
570:= 0} is rarely an ideal,
1361:{\displaystyle 0\times K}
1335:{\displaystyle K\times 0}
1239:{\displaystyle S=\{0,1\}}
1069:prime and non-maximal in
752:{\displaystyle I=p\cap q}
512:Annihilator (ring theory)
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152:project's priority scale
1525:{\displaystyle x\in R.}
1431:then the saturation of
1424:{\displaystyle 0\in S,}
1205:belongs to the other."
1164:{\displaystyle T^{-1}R}
1131:{\displaystyle S^{-1}R}
1058:{\displaystyle S^{-1}p}
1025:{\displaystyle S^{-1}R}
974:{\displaystyle S^{-1}I}
930:{\displaystyle S^{-1}R}
186:Total ring of fractions
109:WikiProject Mathematics
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550:Also btw, {
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1387:-algebras.
247:—Preceding
213:JackSchmidt
123:Mathematics
114:mathematics
70:Mathematics
1594:Categories
1503:for every
1201:such that
356:such that
193:66.92.4.19
1558:D.Lazard
1471:, since
1075:D.Lazard
686:does not
665:D.Lazard
590:Rschwieb
576:Rschwieb
538:Rschwieb
472:D.Lazard
278:unsigned
249:unsigned
1578:Svennik
457:Thomaso
150:on the
41:C-class
1389:EHecky
1372:) but
763:gives
622:= 0}.
47:scale.
1065:with
1582:talk
1562:talk
1393:talk
1376:and
1342:and
1312:and
1246:and
1138:and
1101:and
1079:talk
832:and
727:Let
716:talk
712:Taku
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654:talk
628:talk
594:talk
580:talk
542:talk
520:talk
476:talk
461:talk
286:talk
257:talk
217:talk
197:talk
1451:is
610:; ∀
558:; ∃
532:use
498:; ∃
422:R-p
390:.--
142:Low
1596::
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1482:⋅
1413:∈
1395:)
1353:×
1327:×
1293:×
1203:st
1186:∈
1151:−
1118:−
1081:)
1073:.
1045:−
1012:−
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817:∈
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1257:=
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1234:}
1231:1
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1225:0
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1219:=
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1121:1
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