1466:. What I claim is that the present article has some small deficiencies in describing the history, and some larger problems in describing the actual usages to-day. A reader of the present version should easily get the impression that defining rings as not necessarily unitary was an old habit, which nowadays is abandoned by most writers, while a few have not (yet) changed their outdated mode of using the term "ring". I claim that this is an incorrect impression. Since "ring" is used also in "the older sense" by a number of present authors,
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1101:, despite my preface "as an aside", you presumably misunderstand the spirit of my aside, and seem to be too eager to dismiss a nonstandard perspective while clearly failing to understand it. Do you think that I would even consider misrepresenting the mainstream definitions in WP? I hereby close this aside for discussion (at least by me), as a talk page is not intended for side discussions of this nature. â
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I think that Poonen's main argument is valid and worth to mention - but it indeed is an argument, not a proof. There are also valid arguments for the other side. As for his ring product counterargument, I have a feeling that a careful analysis should show that this more supports the definition witout
1063:
did a logical misstep when he introduced universal algebra? It is not to
Knowledge to change the definition of a variety. The standard definition of a variety, as given in every reliable source, includes 0-ary operations. Moreover, with your suggested change, the theory of varieties would become very
507:
includes all theory of rngs as well as of the specialization to rings (but that article creates the impression that it is restricted to the latter). Given that "ring" has, and still is, used to mean either depending on the author or context, there must be innumerable examples of these. What I would
861:
I agree that the statement creates a false dichotomy. I have tweaked it in a fairly natural way to avoid this. I am not convinced about the claim of the sentence though: "ring" can be used in its restrictive (unital associative) sense while the longer phrase "nonassociative ring" could mean a more
702:
I do not mean that we should change our use "ring" as "unitary ring" in the ring theory articles. This is a sufficiently common usage; and well suited for most of this theory. However, we must distinguish our choices of terminology "for practical purposes" from the description of the various general
334:
Poonen's argument "it is natural to require rings to have a 1" suffers from being an unnatural argument: it unnecessarily posits an infinite axiom system just to provide a context in which to argue that it is "natural" to extend this construction backwards, without arguing for a gain in utility and
1492:
Some of the online encyclopediae you mention look to
Knowledge for their content, so I think it would be better for us to follow not them, but peer-reviewed books written by distinguished authors, say authors who are members of the honorary societies of their respective countries or have won other
1345:
This is the first example of the new 'definitions' in the sixties mentioned by Poonen. Like for the earlier Atiya-McDonald example in our article, this local definition was adapted rather much to the specific demands of a specific work, without making any claim or hint that these local definitions
573:
Following up on the edit I just made removing the Atiyah and MacDonald text as an early example of a unital definition of rings. The text defines rings without a multiplicative identity, before adding that âWe shall consider only rings which are commutative ⊠and have an identity element.â It then
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This does produce another problem, which is that the claim that authors were defining rings as requiring multiplicative identities âas early as the 1960âsâ now has no citation. I donât know enough about the development of the topic to argue for or against that claim, but it seems to me this text
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Ah, my statement re Poonen from three years ago. I'm still tempted to remove the mention of Poonen altogether: saying that in the class of semigroups, loops are prettier does not cut it for declaring the class of semigroups to be incidental/inferior/to be dismissed, and the argument even feels
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this 'older' sense is the one primarily chosen by the three on-line mathematical encyclopediae EOM, MathWorld, and PlanetMath, we should back off a bit here (and possibly restore some older formulations). The reader should be clearly informed that the usage of "ring" as an associative but not
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A minor example: in this article, there is mention of what comes across as a dispute: whether a "ring" should or should not have a '1'. These are simply two classes of object, both valuable, and hence we need to be able to refer to each of them (fortunately settled in this narrow case). The
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As an aside, I would posit that it is a logical misstep in universal algebra to use the 0-ary mapping to posit the existence of an element. It is easy to reformulate the definition of a mapping in this context to show how very sensitive this existence implication is to the interpretation of
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without showing an understanding of the pitfalls of working with triviality. As far as Google
Scholar can tell, this arXiv paper has only incidental citations by two other arXiv papers in the two years since its publication. I suggest that the weight given to Poonen's argument is completely
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Although most modern authors use the term "ring" either as defined here or just without demanding the existence of a unit, there are a few who use the term to refer to more general structures in which there is no requirement for multiplication to be associative. For these authors, every
1350:, published (in the relevant part) 1970. I think that mentions of these two 'local definitions' could be relevant in the historical part. I also think that we should restore a more general mention of the variants already in the lead; e. g., the now outcommented explanatory footnote
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poor, as excluding several important structures (monoids, groups and rings). If fields do not form a variety, it is not because of the existence of 0-ary operations, it is because of the existence of operations that are not everywhere defined (division and multiplicative inverse).
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Do you know of later works by
Grothendieck in which he uses the word "ring" to mean "ring without the requirement of a unit"? That would support the claim that he did not intend for it to be adopted in general. One could ask the same about Atiyah and Macdonald after they wrote
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states that it will gloss âcommutative unitary ringâ as âringâ for the rest of the text. I think that this is substantively different from what the other examples listed do, which is to just define multiplication as associative and possessing an identity.
1394:
Hi
JoergenB, your translation of Grothendieck looks very good to me, though maybe "sauf mention expresse du contraire" (twice) could be translated instead as "unless explicitly stated otherwise", to avoid confusion with the notion of "converse" in
1050:"mapping". An implication in this reformulation is that, for example, that the class of groups is not a variety, just as the fields are not. The class of associative quasigroups is, and it is just the class of groups but for one extra member. â
885:. (I agree with you that that article seem to be given undue weight in our article; I looked at its arXive version, and was not very impressed. However, their main point does not concern infinite products of rings, but rather is an argument for
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necessarily unitary or commutative entity is one of the competing ones in the mathematical world of today; but that we in
Knowledge prefer the one where also unitarity is demanded, and that hence this is what the rest of our article
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Moreover, much of the discussion about unitarity should be placed close to the rest of the definition notes. Since the "not necessarily unitary rings" indeed still abund (as can be seen in EOM, PlanetMath, and MathWorld definitions;
1046:"rng" is an awkward construction, introduced as it was through its spelling. However, I don't see what there is to be done about it (but we should be open to alternatives that are widely used). My example was not perfect, agreed.
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general structure, i.e. with the axiom of associativity dropped. Your comment about 'local definitions' is valid, and EoM's use is hardly a reference for any given usage. I would be inclined to remove the statement altogether. â
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Although most modern authors use the term "ring" as defined here, there are a few who use the term to refer to more general structures in which there is no requirement for multiplication to be associative. For these authors, every
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is intended to recall the example of the ideal of polynomials, commonly referred to as a 'module' or module of forms." This may relate to the etymology, since it is a footnote to the general definition of a (left) ideal denoted
772:
This is part of the mathematical way of thinking, and quite all right. We have rather large possibilities to make local definitions; and we very often need them. I've seen some extremal examples of this (like an article where
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major prizes. My experience is that most of them currently include a 1 in their definition of ring, or at least say early on that their rings will all have a 1 (for the sake of readers who are used to the older convention).
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On the other hand, I note that
Grothendieck's 'local definition' in 1960 also demands a categorical (or universal algebra) property, with mappings ("in general" including inclusions of subrings) respecting the unit element:
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I agree about the general existence of unclearity. (Also see the last point in next section.) Bourbaki (which was pretty influential) does include unitarity in its definition (AlgÚbre I.§8.1), but I think this came a little
781:). I suspect that the prevailing such localised use of "ring" in modern times does include unitarity; but this does not mean that the same authors necessarily employ this local definition "globally", in all other contexts.
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I would suggest a careful review of many articles in ring theory with this in mind, starting with choosing one or more terms to start treating more consistently from a MoS perspective, maybe starting with "ring theory"?
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rings, and when we will speak about a ring without precision, it will be tacitly understood that it concerns a commutative ring. If A is a not necessarily commutative ring, by an A-module we always will mean a
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ought to be adopted in general. Both works certainly were influential; and both may have influenced the decision of the
Bourbakists to change their "official" definition of ring in the "new edition" of their
1259:, et lorsque nous parlerons d'anneau sans préciser, il sera sous-entendu qu'il s'agit d'un anneau commutatif. Si A est un anneau non nécessairement commutatif, par A-module nous entendrons toujours un module
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In several cases (and even regularly in some verb paradigms), vowels in modern
English derived from a svarabhakti being inserted. However, in the modern language, in most such cases, spelling at least gives
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Many times the author or authors of an article specify "local definitions" of some terms, for that article. This may be done rather briefly, by a sentence, or just a parenthesis, in the beginning of the
664:). I do not know if it a mistake or not to neglect those who do not demand unitarity in the quoted paragraph. If it is by intent, then there should be some better support earlier in that section,
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mean that we should change our use "ring" as "unitary ring" in the ring theory articles . There are good reasons for us to fix a consistent terminology in our math articles; and the use of
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717:, while e. g. fields are not. I'm not saying that the one or the other definition is "more natural", though. On the other hand, I'm a bit unhappy with the term "rng", since neither the
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in Welsh is regularly employed for a sound recognised as a vowel in both Welsh and
English, IMHO, this is not a very good example. A more adequate one is offered by the city name
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My reason for addressing you about the finiteness of the axioms for a (unitary) ring was just that I felt a bit unsure of what you meant in your rebuttal of Poonen
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represents an intermediary step in the development of the definition, where lip service is paid to Noetherâs convention, but only unital rings are of interest.
847:, it is not clear to me why you choose to address that comment to me specifically. I have no in-principle issue with your statement about universal algebra.
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like to guard against is creation of incorrect understanding due to the name being used in a way that it is interpreted differently from in the source.
854:", the pronunciation apparently includes an implicit vowel, thus being natural in English, albeit with an unusual spelling (but those abound, such as "
1535:
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and its matrix ring factors are unitary, but the product is taken in the category of not necessarily unitary rings (rngs). Indeed, any (unitary) ring
501:(without qualification) has been settled as being unital and associative when used in WP, it seems to me to be pretty evident that the subject area
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497:, there seems to be some fuzziness in the terminology in this area. Settling further conventions in WP might be helpful. For example, while
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vowel). There are various ways to insert and various qualities of svarabhaktis. One of the most well-established ones is the way the vocalic
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Now, as noted earlier in the section, many authors employ "ring" as not necessarily being unitary ("having a 1"). Actually, so do also the
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changing use of the same term leads to issues with incautious editing if one attaches meaning to terms instead of the other way around.
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except not demanding the uniqueness": Indeed, a "rng" is "a ring, except not necessarily containing a 'one'" (which may be written
779:(but in the case with the non-commutative addition, if I had been the referee, I might have suggested some changes of terminology
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in general are employed as vowels in modern English, while our texts should be pronounceable; but surely there are alternatives.)
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in "a ring" was not necessarily additive); as long as the local definitions employed are clarified, this yields no real trouble
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It is true that many foreign words (including some names) borrowed into English usually retain their original spelling (as the
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I can see from the lack of response that this has effectively no support, so I will consider myself outvoted. â
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which are not ideals were introduced as "syzygy modules" by Hilbert in the same famous paper, for the proof of
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under multiplication; that is, do not require that there be a multiplicative identity (1). See the section
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370:: Noether has a footnote (courtesy of Google translate): "Ideals are denoted with capital German letters.
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Thanks for your comments! I'm sorry if I didn't express myself clear enough. As I wrote above, I do
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until a consensus is reached, and readers of this page are welcome to contribute to the discussion.
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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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are reasonably easy to accomodate in the modern English language, I wouldn't state the same for
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I note that unitary rings are "definable with a finite number of axioms" within the context of
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A good place to discuss Knowledge's convention that rings have a 1 might be the talk page for
1375:), we should not pretend that this just is a lingering but essentially historical definition.
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hint about the modern pronunciation of the resulting vowel sound. Introducing a new word like
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different from both 0 and 1 may be exhibited as a sum or a product of two unitary subrings
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In fact, before Hilbert, ideals that were considered were only those of number theory and
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Thanks â that already is helpful. Also nice to know that I'm not totally out on a limb
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noteworthy alternative definition around demands unitarity but not associativity:
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of the most common extant usages. (On the other hand, I'm unhappy with the term
1272:ââA. Grothendieck, ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique, Chapitre 0 (PrĂ©liminaires),
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Institute des hautes études scientifiques, 1960, Publications mathématiques, N 4
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is represented in modern Indian usage, namely, by inserting a svarabhakti
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as including unitarity and assiciativity but not commutativity indeed is
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here originally standing for the diphthong in the modern English pronoun
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At present, there is a paragraph which implicitly seems to indicate that
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In this article, all rings are commutative, unitary, and noetherian.
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indeed respects units. Now, this is true; but IMHO discarding
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I found a reference, which I added to the history section of
948:(or, in British English, even a complete replacement of the
858:"). Do we have a better term to use here? I don't see one.
459:. The first use of ideals outside number theory seems to be
952:), but also gives the Czech pronunciation (with a thrilled
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Tous les anneaux considérés dans ce Traité posséderont un
1215:{\displaystyle B\simeq B\times 0\subset B\times C\simeq A}
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Rather unclear formulation implicitly including unitarity
1322:; the ring homomorphisms are always assumed supposed to
1247:; les homomorphismes d'anneaux seront toujours supposés
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our choice of meaning of "ring", but our report of the
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unitarity. So, forinstance, a von Neumann regular ring
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Else, the paragraph should be modified, forinstance to
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be any (unital but not necessarily commutative) ring.
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A discussion is taking place to address the redirect
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module, except explicit mention of the converse.")
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1314:"All rings considered in this Work will have a
1449:Knowledge:Manual of Style/Mathematics#Algebra
1408:Knowledge:Manual of Style/Mathematics#Algebra
878:Thanks for fixing the false dichotomy issue!
275:This page has archives. Sections older than
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1249:transformer l'élément unité en élément unité
1129:is a product of matrix algebras, where both
912:in Roman numerals). However, while words as
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674:in the ring definition itself
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167:and see a list of open tasks.
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838:
837:
819:
800:
799:
795:
794:
793:
770:
769:
768:
755:
743:
732:
731:
727:
721:nor the nasal
696:
695:
694:
693:
642:
641:
640:
639:
618:
615:
614:
613:
570:
567:
566:
565:
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563:
543:
542:
493:As I noted at
490:
487:
486:
485:
484:
483:
406:
381:
361:
358:
357:
356:
331:
328:
299:
293:
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274:
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232:
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191:
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165:the discussion
152:
151:
135:
123:
122:
114:
102:
101:
95:
73:
59:
58:
55:
54:
51:
44:
36:
35:
32:
29:
25:
24:
13:
10:
9:
6:
4:
3:
2:
1578:
1567:
1564:
1562:
1559:
1557:
1554:
1552:
1549:
1547:
1544:
1542:
1539:
1537:
1534:
1532:
1529:
1527:
1524:
1523:
1521:
1504:
1500:
1496:
1495:Ebony Jackson
1491:
1488:
1487:
1486:
1482:
1478:
1474:
1469:
1465:
1461:
1457:
1453:
1450:
1446:
1442:
1438:
1434:
1428:
1427:Ebony Jackson
1423:
1422:
1421:
1417:
1413:
1412:Ebony Jackson
1409:
1405:
1402:
1397:
1393:
1392:
1391:
1390:
1389:
1388:
1387:
1386:
1382:
1378:
1374:
1363:
1359:
1355:
1354:
1353:
1352:
1351:
1349:
1338:
1333:
1329:
1325:
1321:
1317:
1313:
1312:
1311:
1310:
1309:
1305:
1293:
1288:
1284:
1280:
1275:
1269:
1262:
1258:
1254:
1250:
1246:
1242:
1241:élement unité
1238:
1234:
1231:
1227:
1225:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1171:
1167:
1163:
1159:
1155:
1151:
1147:
1143:
1140:
1136:
1132:
1128:
1122:
1107:
1104:
1100:
1097:
1096:
1095:
1094:
1093:
1092:
1091:
1090:
1089:
1088:
1087:
1086:
1075:
1071:
1067:
1062:
1058:
1057:
1056:
1053:
1048:
1045:
1041:
1040:
1039:
1038:
1037:
1036:
1035:
1034:
1027:
1023:
1019:
1015:
1011:
1007:
1003:
998:
995:
991:
987:
983:
979:
975:
971:
967:
963:
959:
955:
951:
947:
943:
939:
935:
932:
928:
927:
923:
919:
915:
911:
907:
903:
899:
895:
892:
888:
884:
880:
879:
875:
870:
869:
868:
865:
860:
857:
853:
849:
846:
843:
842:
833:
829:
823:
820:
815:
811:
805:
802:
798:
792:
788:
784:
776:
771:
767:
753:
744:
742:
739:
736:
735:
734:
733:
728:
720:
716:
710:
701:
700:
699:
698:Two remarks:
692:
690:
684:
683:
682:
681:
680:
677:
675:
671:
667:
663:
659:
655:
651:
647:
638:
636:
630:
629:
628:
627:
626:
624:
616:
612:
608:
604:
597:
592:
591:
590:
589:
585:
581:
575:
568:
562:
559:
547:
546:
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544:
541:
537:
533:
532:Ebony Jackson
529:
525:
524:
523:
522:
519:
513:
509:
506:
505:
500:
496:
488:
482:
478:
474:
470:
466:
462:
458:
454:
450:
449:
448:
444:
440:
439:Ebony Jackson
436:
432:
431:
430:
429:
426:
422:
368:
359:
355:
352:
348:
347:
346:
345:
342:
338:
330:Undue weight?
329:
327:
326:
322:
318:
314:
310:
305:
298:
294:
292:
283:
278:
273:
272:
258:
257:
254:
253:
249:
245:
241:
238:
234:
233:
229:
224:
219:
218:
215:
200:
196:
190:
187:
186:
183:
166:
162:
158:
157:
149:
143:
138:
136:
133:
129:
128:
124:
118:
115:
112:
108:
103:
99:
93:
85:
84:
74:
70:
65:
64:
52:
50:
49:
45:
42:
38:
37:
33:
30:
27:
26:
21:
17:
1467:
1463:
1459:
1444:
1440:
1436:
1432:
1400:
1372:
1369:
1347:
1344:
1336:
1331:
1327:
1323:
1319:
1316:unit element
1315:
1299:
1291:
1286:
1282:
1278:
1273:
1260:
1256:
1252:
1248:
1244:
1240:
1228:
1223:
1169:
1165:
1164:onto either
1161:
1157:
1153:
1149:
1145:
1141:
1134:
1130:
1126:
1123:
1120:
1016:suggestion.
1013:
1009:
1005:
1001:
993:
985:
981:
977:
973:
969:
961:
958:phonological
953:
949:
937:
933:
921:
917:
913:
909:
901:
897:
882:
831:
822:
813:
804:
796:
775:the addition
774:
745:
740:
737:
718:
697:
691:is a "ring".
685:
678:
673:
643:
637:is a "ring".
631:
622:
620:
576:
572:
514:
510:
502:
498:
492:
364:Prompted by
363:
333:
301:
290:
276:
235:
227:
214:
195:Top-priority
194:
154:
120:Topâpriority
98:WikiProjects
81:
46:
1332:commutative
1257:commutatifs
1010:pseudo-ring
946:svarabhakti
703:practices.
528:Ring theory
504:Ring theory
489:Terminology
309:Ring action
170:Mathematics
161:mathematics
117:Mathematics
48:Peer review
1520:Categories
1373:vide supra
1139:idempotent
972:after the
891:semigroups
797:References
658:PlanetMath
1358:semigroup
1245:unitaires
906:universal
650:MathWorld
86:is rated
1477:JoergenB
1377:JoergenB
1304:D.Lazard
1261:Ă gauche
1099:D.Lazard
1066:D.Lazard
1018:JoergenB
966:Sanskrit
918:morphism
845:JoergenB
783:JoergenB
730:article:
623:the only
603:JoergenB
473:D.Lazard
337:WP:UNDUE
317:D.Lazard
277:365 days
228:Archives
53:Reviewed
1348:AlgĂšbre
1320:unitary
1103:Quondum
1052:Quondum
990:Rigveda
978:áčgwaida
896:As for
887:monoids
874:Quondum
864:Quondum
709:Quondum
689:algebra
656:), and
635:algebra
558:Quondum
518:Quondum
465:modules
425:Quondum
351:Quondum
341:Quondum
197:on the
88:B-class
31:Process
1395:maths.
1156:and 1-
980:(with
914:versal
902:versal
601:later.
596:Lnkov1
580:Lnkov1
94:scale.
34:Result
1403:book.
1401:their
931:Welsh
883:supra
435:ideal
237:Index
75:This
1499:talk
1481:talk
1473:ring
1456:ring
1437:ring
1416:talk
1381:talk
1337:left
1148:and
1070:talk
1022:talk
1002:some
994:ring
942:Brno
850:On "
787:talk
746:Let
607:talk
584:talk
536:talk
499:ring
477:talk
443:talk
367:this
321:talk
28:Date
1468:and
1460:not
1445:rng
1441:one
1433:not
1290:);
1224:not
1168:or
1006:rng
964:in
934:cwm
922:rng
916:or
898:rng
856:cwm
852:rng
646:EOM
189:Top
1522::
1501:)
1483:)
1451:.)
1418:)
1383:)
1266:â
1236:â
1207:â
1201:Ă
1195:â
1189:Ă
1183:â
1072:)
1024:)
982:ai
893:.)
830:.
812:.
789:)
741:or
676:.
648:,
609:)
586:)
538:)
479:)
471:.
445:)
437:.
323:)
250:,
246:,
242:,
1497:(
1479:(
1429::
1425:@
1414:(
1379:(
1306:,
1302:@
1287:1
1285:.
1283:0
1281:.
1279:1
1276:(
1210:A
1204:C
1198:B
1192:0
1186:B
1180:B
1170:C
1166:B
1162:A
1158:i
1154:i
1150:C
1146:B
1142:i
1135:A
1131:R
1127:R
1068:(
1020:(
986:I
974:r
970:i
962:r
954:r
950:r
938:w
924:.
910:i
876::
872:@
834:.
816:.
785:(
754:A
723:Ć
719:r
711::
707:@
705:(
660:(
652:(
605:(
598::
594:@
582:(
556:â
534:(
516:â
475:(
441:(
405:M
380:M
319:(
252:4
248:3
244:2
240:1
201:.
100::
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