253:
243:
222:
533:
convergent power series is a unit or not, you only have to check it's inverse is analytic at 0 or not. Also, some adding some concrete familiar example is helpful for readers. Theoretically, maybe formal power series are easier to handle, but surely more readers are familiar with convergent power series. If the imprecision of the definition is what bothers you than that's easy to fix, I think. --
191:
1826:
705:
as the preferred notation for the unit group. I think that this more compact notation, introduced by Weil, is more common in the current mathematical literature, especially in algebraic number theory, which is probably the field in which unit groups are used most extensively. Another example: It is
618:
basically discusses an algebra of convergent power series. (since they start with continuous functions, you have to consider germs, not functions, though.) Since a germ of a holomorphic function is a power series, I still think "the algebra of convergent power series around 0" is a simple and typical
570:
I didn't use the term "analytic functions" because then the example would be too simple to be interesting. (I also admit since the article now at least mentions about a local ring, this example ceases to be interesting, because you can say this algebra is just a local ring.) Anyway, the point is I
474:
Oh, I thought "convergent" simplifies the idea. In my view, a formal power series is a generalization of a convergent power series. I thought the example was interesting because this is exactly how I learned about units and non-units. That's why I was surprised to find that the article didn't mean
532:
By saying "at 0" I thought that would mean "in some neighborhood of 0" (so disk or interval, etc.) Maybe "around 0" or "near 0" is better wording? (I admit this might be a little bit imprecise.) Also, I didn't think formal power series or rational functions are easier to hand, because to see if a
655:
But "a matrix with all diagonal elements equal to one, and all other elements equal to zero" is the very meaning that was intended earlier, isn't it? If so, suggest removing "more usually". I don't know enough about ring theory to be sure, though, so I won't make the change myself.
1302:
By the way, I am not sure using the term "group scheme" is a good idea. Obviously that's what I had in mind but the article can and should be written to avoid the reference to scheme theory (the readers with an appropriate background can see we are talking about group scheme). --
1751:
I believe that the terms "unit" and "invertible element" are synonymous in ring theory ("unit" in the sense of this article of course, not the multiplicative identity element) – if so then we should mention in the introduction that units also go by this name.
556:
know they're the same, but the fact that the reciprocal of an analytic function, non-zero at the specified point, is also analytic, is probably closer to "common sense" then that the inverse of a convergent power series is convergent. —
432:
In an algebra of convergent power series at the origin, units are precisely those who do not vanish at the origin. The non-units (i.e., those vanish at the origin) then form a unique maximal idea, (thus, this algebra is a local
650:"unity", and say that ''R'' is a "ring with unity" rather than "ring with a unit". Note also that the term '']'' more usually denotes a matrix with all diagonal elements equal to one, and all other elements equal to zero.)
1122:
Ah, I see. I think it matters of the perspective (a choice of the notation can certainly reflect that): in some theoretical sense, I think it is important (and worth mentioned in some form) that, for a commutative ring,
153:
1405:
U(R). With GL_1, it is not so much of an issue, since it is reasonable to take the point of view that GL_1(R) is defined to be the group of invertible 1 x 1 matrices. I would probably omit the mention of
1755:(note: "invertible element" is unambiguous in the context of ring theory since every element of a ring is additivity invertible, so clearly "invertible" must be refering to multiplicative invertibility)
1084:
1008:
493:? On an interval containing 0? On a disk in the complex plane containing 0? (Well, those are the same, and it's well-defined.) I guess we differ on the appropriate level of abstraction. "
1227:
309:
1838:
1485:
1046:
943:
850:
771:
1634:
1597:
1667:
1544:
1515:
1433:
1402:
1373:
1344:
1297:
1268:
1183:
1150:
897:
1106:
798:
147:
614:
Locally, of course (or depending on your definition of analyticity. I usually start with holomorphic = analytic.). Anyway, we are getting off-track. I noticed
1871:
299:
79:
44:
1866:
437:
Is this mathematically incorrect? (I admit maybe I'm not understanding terminology correct, but I don't think the example itself is wrong.) --
275:
1669:
seems to make sense). Also, mentioning this representability fact should also help somehow put group ring construction somehow in context. --
513:, but it's easier to see in the case of formal power series or rational functions than for convergent power series or analytic functions. —
85:
1834:
1818:
648:
of the ring, in expressions like ''ring with a unit'' or '']'', and also e.g. '']''. (For this reason, some authors call 1<sub: -->
1487:
is probably not common outside algebraic geometry". I suppose the question is: what is the common notation for the functor R --: -->
266:
227:
1241:(theoretically important, since, for example, the deformation theory should be able to give a sense to a group functor in between
773:, which is probably not common outside algebraic geometry, though. For a non-commutative ring, I am not too sure if the notation
1546:
isn't really have a discussion on a particular group scheme but the fact that a unit element corresponds to a ring homomorphism
168:
99:
30:
135:
104:
20:
1346:
is used more commonly for the group scheme that represents the functor, not so much the functor itself, so I'd say that if
74:
1055:
202:
1723:
I agree with this completely and support changing the notation in the article, the biggest reason being simply that U(
65:
962:
462:
has the property that the units are exactly those with a unit constanrt term. "Convergent" just adds confusion. —
1192:
129:
414:
190:
125:
1806:
1698:
1440:
1113:
861:
725:
502:
109:
1760:
1736:
1674:
1308:
1234:
950:
805:
624:
602:
580:
561:
538:
517:
480:
466:
442:
1452:
1013:
910:
817:
738:
208:
175:
1756:
1732:
252:
1602:
1549:
1841:
until a consensus is reached, and readers of this page are welcome to contribute to the discussion.
1643:
1520:
1491:
1409:
1378:
1349:
1320:
1273:
1244:
1159:
1126:
873:
661:
410:
376:
the "(ring theory)" in the second one arose historically as a disambiguation from other meanings of
494:
455:
161:
55:
274:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
1830:
1802:
1694:
1436:
1109:
1089:
857:
721:
358:
336:
258:
70:
24:
776:
242:
221:
1670:
1304:
1230:
946:
801:
676:
620:
599:
576:
558:
549:
534:
514:
506:
498:
476:
463:
438:
384:
redirects here, maybe a bit misleadingly - since "unit (algebra)" could mean both, the 1 of a
141:
51:
1783:
646:
Unfortunately, the term ''unit'' is also used to refer to the identity element 1<sub: -->
657:
389:
388:
algebra, and any unit of an algebra, seen as a ring (and, also, the algebraic notion of
381:
348:
1860:
1848:
800:
is that common (but I don't know much about the non-commutative ring literature.) --
401:
1108:
every time one wanted to refer to the field of rational numbers. Nobody does that!
575:
example. If you disagree on this, then I'm fine with the removal of the example. --
672:
331:
Even though both articles are a bit stub-like, care should be taken before merging
945:= the group of unit elements. Maybe you're thinking of a different definition? --
1779:
409:
I disagree with merger also, these articles are talking about different things.
271:
615:
510:
248:
1801:
is its ring of integers. I fixed this (and a bunch of other little things!)
1517:
seems like a common notation for that functor. Also, the point of mentioning
1852:
1839:
Knowledge:Redirects for discussion/Log/2022 December 10 § Invertible element
1810:
1787:
1764:
1740:
1702:
1678:
1444:
1312:
1238:
1117:
954:
899:
refers to the multiplicative group and the unit group of a commutative ring
865:
809:
729:
680:
665:
628:
605:
584:
564:
542:
520:
484:
469:
446:
418:
404:
366:
344:
332:
856:
for the unit group. It is just a group that happens to be the unit group.
1599:. This seems important; it contrasts to the fact that a ring homomorphism
1843:
671:
You're right. That last sentence is redundant. I'm going to remove it.
397:
1825:
1488:
U(R)? I don't think that's U (right?). At least in algebraic geometry,
385:
380:- it could have been chosen to be "(algebra)" quite as well - b.t.w.
377:
1829:
An editor has identified a potential problem with the redirect
1189:
of the ring. But I think I get your point is that the notation
184:
15:
1375:
is going to be mentioned at all, it is clearer to say that
1404:
is the group scheme that represents the functor R --: -->
1778:
in the example with integers? There is no reference to.
959:
Sorry for not being clearer. It is certainly true that
598:, but it's stated in the analytic function article. —
1449:
That's what I wanted to mean when I say "the notation
720:
as a notation for the set of nonzero complex numbers.
160:
1646:
1605:
1552:
1523:
1494:
1455:
1412:
1381:
1352:
1323:
1276:
1247:
1195:
1162:
1129:
1092:
1058:
1016:
965:
913:
876:
820:
779:
741:
351:
are different objects and merit IMHO seperate entries
270:, a collaborative effort to improve the coverage of
1079:{\displaystyle \operatorname {Frac} (\mathbb {Z} )}
619:example. What makes it not a "typical" example? --
1661:
1628:
1591:
1538:
1509:
1479:
1427:
1396:
1367:
1338:
1291:
1262:
1221:
1177:
1144:
1100:
1078:
1040:
1002:
937:
891:
844:
814:I would say that not even algebraic geometers use
792:
765:
454:It doesn't seem that interesting an example. The
33:for general discussion of the article's subject.
1003:{\displaystyle \mathbb {G} _{m}(R)=R^{\times }}
571:didn't think it was confusing, but rather very
396:Personally, I'm rather against this merging. —
365:unit (= 1) of the (unital) ring (!!!) - while
1222:{\displaystyle \mathbb {G} _{m}(\mathbb {C} )}
552:rather than convergent power series. I mean,
903:is the same thing as multiplicative group of
505:of (or disk containing) 0, or the subring of
174:
8:
1683:I would refer to it simply as " the functor
1435:; that is straying from the topic of units.
188:
216:
1653:
1649:
1648:
1645:
1607:
1606:
1604:
1571:
1554:
1553:
1551:
1530:
1526:
1525:
1522:
1501:
1497:
1496:
1493:
1462:
1458:
1457:
1454:
1419:
1415:
1414:
1411:
1388:
1384:
1383:
1380:
1359:
1355:
1354:
1351:
1330:
1326:
1325:
1322:
1283:
1279:
1278:
1275:
1254:
1250:
1249:
1246:
1212:
1211:
1202:
1198:
1197:
1194:
1169:
1165:
1164:
1161:
1136:
1132:
1131:
1128:
1094:
1093:
1091:
1069:
1068:
1057:
1052:for the unit group would be like writing
1023:
1019:
1018:
1015:
994:
972:
968:
967:
964:
920:
916:
915:
912:
883:
879:
878:
875:
827:
823:
822:
819:
784:
778:
748:
744:
743:
740:
218:
1793:Thank you for pointing this out. The
7:
590:But convergent power series are the
264:This article is within the scope of
1480:{\displaystyle \mathbb {G} _{m}(R)}
1041:{\displaystyle \mathbb {G} _{m}(R)}
938:{\displaystyle \mathbb {G} _{m}(R)}
907:; i.e., so, for me, by definition,
845:{\displaystyle \mathbb {G} _{m}(R)}
766:{\displaystyle \mathbb {G} _{m}(R)}
497:", or "convergent power series" = "
361:is any invertible element, and not
207:It is of interest to the following
23:for discussing improvements to the
1185:the underlying abelian group, the
373:unit (neutral for multiplication).
14:
1872:Mid-priority mathematics articles
1629:{\displaystyle \mathbb {Z} \to R}
1592:{\displaystyle \mathbb {Z} \to R}
735:Well, there is also the notation
594:as analytic functions. It's not
284:Knowledge:WikiProject Mathematics
1867:Start-Class mathematics articles
1837:. This discussion will occur at
1824:
1662:{\displaystyle \mathbb {G} _{a}}
1539:{\displaystyle \mathbb {G} _{m}}
1510:{\displaystyle \mathbb {G} _{m}}
1428:{\displaystyle \mathbb {G} _{a}}
1397:{\displaystyle \mathbb {G} _{m}}
1368:{\displaystyle \mathbb {G} _{m}}
1339:{\displaystyle \mathbb {G} _{m}}
1292:{\displaystyle \mathbb {G} _{a}}
1263:{\displaystyle \mathbb {G} _{m}}
1178:{\displaystyle \mathbb {G} _{a}}
1145:{\displaystyle \mathbb {G} _{m}}
892:{\displaystyle \mathbb {G} _{m}}
509:defined at 0. They are are all
287:Template:WikiProject Mathematics
251:
241:
220:
189:
45:Click here to start a new topic.
1817:"Invertible element" listed at
304:This article has been rated as
1620:
1617:
1611:
1583:
1580:
1558:
1474:
1468:
1216:
1208:
1073:
1065:
1035:
1029:
984:
978:
932:
926:
839:
833:
760:
754:
1:
1853:16:21, 10 December 2022 (UTC)
1811:18:37, 29 December 2021 (UTC)
1788:12:37, 23 December 2021 (UTC)
1636:corresponds to an element of
730:18:35, 27 December 2020 (UTC)
681:03:01, 2 September 2009 (UTC)
278:and see a list of open tasks.
42:Put new text under old text.
1101:{\displaystyle \mathbb {Q} }
870:I am confused; the notation
1703:17:16, 4 January 2021 (UTC)
1679:09:08, 4 January 2021 (UTC)
1445:06:56, 4 January 2021 (UTC)
1313:05:50, 4 January 2021 (UTC)
1239:05:43, 4 January 2021 (UTC)
1118:06:00, 3 January 2021 (UTC)
955:05:50, 3 January 2021 (UTC)
866:05:46, 3 January 2021 (UTC)
810:05:06, 3 January 2021 (UTC)
793:{\displaystyle R^{\times }}
50:New to Knowledge? Welcome!
1888:
1010:. But I think that using
666:00:02, 16 March 2009 (UTC)
419:03:56, 10 March 2006 (UTC)
405:03:49, 10 March 2006 (UTC)
354:notice in particular that
1765:11:23, 23 July 2021 (UTC)
1741:11:18, 23 July 2021 (UTC)
303:
236:
215:
80:Be welcoming to newcomers
1819:Redirects for discussion
706:much more common to see
691:We could consider using
629:23:10, 30 May 2008 (UTC)
606:22:22, 30 May 2008 (UTC)
585:21:56, 30 May 2008 (UTC)
565:21:39, 30 May 2008 (UTC)
543:21:25, 30 May 2008 (UTC)
521:13:56, 30 May 2008 (UTC)
485:06:25, 30 May 2008 (UTC)
470:21:54, 29 May 2008 (UTC)
447:21:18, 29 May 2008 (UTC)
310:project's priority scale
1833:and has thus listed it
1797:is a number field, and
687:Notation for unit group
641:The introduction says:
267:WikiProject Mathematics
1727:) is less common than
1663:
1630:
1593:
1540:
1511:
1481:
1429:
1398:
1369:
1340:
1293:
1264:
1223:
1179:
1146:
1102:
1080:
1042:
1004:
939:
893:
846:
794:
767:
475:about this at all. --
197:This article is rated
75:avoid personal attacks
1664:
1631:
1594:
1541:
1512:
1482:
1430:
1399:
1370:
1341:
1317:I think the notation
1294:
1265:
1224:
1180:
1147:
1103:
1081:
1043:
1005:
940:
894:
847:
795:
768:
100:Neutral point of view
1747:"Invertible element"
1644:
1603:
1550:
1521:
1492:
1453:
1410:
1379:
1350:
1321:
1274:
1245:
1229:does look weird. --
1193:
1160:
1154:multiplicative group
1127:
1090:
1056:
1014:
963:
911:
874:
818:
777:
739:
649:'''R'''</sub: -->
637:Misleading Statement
428:About this example:
424:Power series example
392:for multiplication).
290:mathematics articles
105:No original research
1156:of the ring, while
495:Formal power series
456:formal power series
1831:Invertible element
1659:
1626:
1589:
1536:
1507:
1477:
1425:
1394:
1365:
1336:
1289:
1260:
1219:
1175:
1142:
1098:
1076:
1038:
1000:
935:
889:
842:
790:
763:
647:''R''</sub: -->
550:analytic functions
507:rational functions
499:analytic functions
359:unit (ring theory)
337:unit (ring theory)
259:Mathematics portal
203:content assessment
86:dispute resolution
47:
25:Unit (ring theory)
324:
323:
320:
319:
316:
315:
183:
182:
66:Assume good faith
43:
1879:
1851:
1846:
1828:
1692:
1668:
1666:
1665:
1660:
1658:
1657:
1652:
1635:
1633:
1632:
1627:
1610:
1598:
1596:
1595:
1590:
1579:
1578:
1557:
1545:
1543:
1542:
1537:
1535:
1534:
1529:
1516:
1514:
1513:
1508:
1506:
1505:
1500:
1486:
1484:
1483:
1478:
1467:
1466:
1461:
1434:
1432:
1431:
1426:
1424:
1423:
1418:
1403:
1401:
1400:
1395:
1393:
1392:
1387:
1374:
1372:
1371:
1366:
1364:
1363:
1358:
1345:
1343:
1342:
1337:
1335:
1334:
1329:
1298:
1296:
1295:
1290:
1288:
1287:
1282:
1269:
1267:
1266:
1261:
1259:
1258:
1253:
1228:
1226:
1225:
1220:
1215:
1207:
1206:
1201:
1184:
1182:
1181:
1176:
1174:
1173:
1168:
1151:
1149:
1148:
1143:
1141:
1140:
1135:
1107:
1105:
1104:
1099:
1097:
1085:
1083:
1082:
1077:
1072:
1047:
1045:
1044:
1039:
1028:
1027:
1022:
1009:
1007:
1006:
1001:
999:
998:
977:
976:
971:
944:
942:
941:
936:
925:
924:
919:
898:
896:
895:
890:
888:
887:
882:
851:
849:
848:
843:
832:
831:
826:
799:
797:
796:
791:
789:
788:
772:
770:
769:
764:
753:
752:
747:
719:
711:
704:
696:
292:
291:
288:
285:
282:
261:
256:
255:
245:
238:
237:
232:
224:
217:
200:
194:
193:
185:
179:
178:
164:
95:Article policies
16:
1887:
1886:
1882:
1881:
1880:
1878:
1877:
1876:
1857:
1856:
1847:
1842:
1822:
1772:
1749:
1684:
1647:
1642:
1641:
1640:(so mentioning
1601:
1600:
1567:
1548:
1547:
1524:
1519:
1518:
1495:
1490:
1489:
1456:
1451:
1450:
1413:
1408:
1407:
1382:
1377:
1376:
1353:
1348:
1347:
1324:
1319:
1318:
1277:
1272:
1271:
1248:
1243:
1242:
1196:
1191:
1190:
1163:
1158:
1157:
1130:
1125:
1124:
1088:
1087:
1054:
1053:
1017:
1012:
1011:
990:
966:
961:
960:
914:
909:
908:
877:
872:
871:
821:
816:
815:
780:
775:
774:
742:
737:
736:
713:
707:
698:
692:
689:
639:
426:
411:Oleg Alexandrov
390:neutral element
343:on one hand, a
329:
289:
286:
283:
280:
279:
257:
250:
230:
201:on Knowledge's
198:
121:
116:
115:
114:
91:
61:
12:
11:
5:
1885:
1883:
1875:
1874:
1869:
1859:
1858:
1835:for discussion
1821:
1815:
1814:
1813:
1771:
1768:
1748:
1745:
1744:
1743:
1720:
1719:
1718:
1717:
1716:
1715:
1714:
1713:
1712:
1711:
1710:
1709:
1708:
1707:
1706:
1705:
1656:
1651:
1625:
1622:
1619:
1616:
1613:
1609:
1588:
1585:
1582:
1577:
1574:
1570:
1566:
1563:
1560:
1556:
1533:
1528:
1504:
1499:
1476:
1473:
1470:
1465:
1460:
1422:
1417:
1391:
1386:
1362:
1357:
1333:
1328:
1300:
1286:
1281:
1257:
1252:
1218:
1214:
1210:
1205:
1200:
1187:additive group
1172:
1167:
1139:
1134:
1096:
1075:
1071:
1067:
1064:
1061:
1037:
1034:
1031:
1026:
1021:
997:
993:
989:
986:
983:
980:
975:
970:
934:
931:
928:
923:
918:
886:
881:
841:
838:
835:
830:
825:
787:
783:
762:
759:
756:
751:
746:
688:
685:
684:
683:
653:
652:
638:
635:
634:
633:
632:
631:
609:
608:
568:
567:
530:
529:
528:
527:
526:
525:
524:
523:
458:ring over any
435:
434:
425:
422:
394:
393:
382:unit (algebra)
374:
352:
349:unit of a ring
328:
325:
322:
321:
318:
317:
314:
313:
302:
296:
295:
293:
276:the discussion
263:
262:
246:
234:
233:
225:
213:
212:
206:
195:
181:
180:
118:
117:
113:
112:
107:
102:
93:
92:
90:
89:
82:
77:
68:
62:
60:
59:
48:
39:
38:
35:
34:
28:
13:
10:
9:
6:
4:
3:
2:
1884:
1873:
1870:
1868:
1865:
1864:
1862:
1855:
1854:
1850:
1845:
1840:
1836:
1832:
1827:
1820:
1816:
1812:
1808:
1804:
1803:Ebony Jackson
1800:
1796:
1792:
1791:
1790:
1789:
1785:
1781:
1777:
1769:
1767:
1766:
1762:
1758:
1753:
1746:
1742:
1738:
1734:
1730:
1726:
1722:
1721:
1704:
1700:
1696:
1695:Ebony Jackson
1691:
1687:
1682:
1681:
1680:
1676:
1672:
1654:
1639:
1623:
1614:
1586:
1575:
1572:
1568:
1564:
1561:
1531:
1502:
1471:
1463:
1448:
1447:
1446:
1442:
1438:
1437:Ebony Jackson
1420:
1389:
1360:
1331:
1316:
1315:
1314:
1310:
1306:
1301:
1284:
1255:
1240:
1236:
1232:
1203:
1188:
1170:
1155:
1137:
1121:
1120:
1119:
1115:
1111:
1110:Ebony Jackson
1062:
1059:
1051:
1032:
1024:
995:
991:
987:
981:
973:
958:
957:
956:
952:
948:
929:
921:
906:
902:
884:
869:
868:
867:
863:
859:
858:Ebony Jackson
855:
836:
828:
813:
812:
811:
807:
803:
785:
781:
757:
749:
734:
733:
732:
731:
727:
723:
722:Ebony Jackson
717:
710:
702:
695:
686:
682:
678:
674:
670:
669:
668:
667:
663:
659:
651:
644:
643:
642:
636:
630:
626:
622:
617:
613:
612:
611:
610:
607:
604:
601:
597:
593:
589:
588:
587:
586:
582:
578:
574:
566:
563:
560:
555:
551:
548:Perhaps just
547:
546:
545:
544:
540:
536:
522:
519:
516:
512:
508:
504:
501:defined in a
500:
496:
492:
488:
487:
486:
482:
478:
473:
472:
471:
468:
465:
461:
457:
453:
452:
451:
450:
449:
448:
444:
440:
431:
430:
429:
423:
421:
420:
416:
412:
407:
406:
403:
399:
391:
387:
383:
379:
375:
372:
368:
364:
360:
357:
353:
350:
346:
342:
341:
340:
338:
334:
326:
311:
307:
301:
298:
297:
294:
277:
273:
269:
268:
260:
254:
249:
247:
244:
240:
239:
235:
229:
226:
223:
219:
214:
210:
204:
196:
192:
187:
186:
177:
173:
170:
167:
163:
159:
155:
152:
149:
146:
143:
140:
137:
134:
131:
127:
124:
123:Find sources:
120:
119:
111:
110:Verifiability
108:
106:
103:
101:
98:
97:
96:
87:
83:
81:
78:
76:
72:
69:
67:
64:
63:
57:
53:
52:Learn to edit
49:
46:
41:
40:
37:
36:
32:
26:
22:
18:
17:
1823:
1798:
1794:
1775:
1773:
1757:Joel Brennan
1754:
1750:
1733:Joel Brennan
1728:
1724:
1689:
1685:
1637:
1186:
1153:
1049:
904:
900:
853:
715:
708:
700:
693:
690:
654:
645:
640:
600:Arthur Rubin
595:
591:
573:prototypical
572:
569:
559:Arthur Rubin
553:
531:
515:Arthur Rubin
503:neighborhood
490:
464:Arthur Rubin
459:
436:
427:
408:
395:
370:
362:
355:
330:
306:Mid-priority
305:
265:
231:Mid‑priority
209:WikiProjects
171:
165:
157:
150:
144:
138:
132:
122:
94:
19:This is the
1086:instead of
712:instead of
697:instead of
511:local rings
489:Convergent
281:Mathematics
272:mathematics
228:Mathematics
199:Start-class
148:free images
31:not a forum
1861:Categories
1152:gives the
658:BlueGuy213
616:Local ring
369:refers to
367:unit ring
345:unit ring
333:unit ring
88:if needed
71:Be polite
21:talk page
1774:What is
1050:notation
854:notation
327:Untitled
56:get help
29:This is
27:article.
673:Rckrone
596:obvious
308:on the
154:WP refs
142:scholar
1780:Madyno
603:(talk)
562:(talk)
518:(talk)
467:(talk)
433:ring.)
386:unital
347:and a
205:scale.
126:Google
1048:as a
852:as a
491:where
169:JSTOR
130:books
84:Seek
1807:talk
1784:talk
1761:talk
1737:talk
1699:talk
1675:talk
1671:Taku
1441:talk
1309:talk
1305:Taku
1270:and
1235:talk
1231:Taku
1114:talk
1060:Frac
951:talk
947:Taku
862:talk
806:talk
802:Taku
726:talk
677:talk
662:talk
625:talk
621:Taku
592:same
581:talk
577:Taku
539:talk
535:Taku
481:talk
477:Taku
460:ring
443:talk
439:Taku
415:talk
402:Talk
378:unit
335:and
162:FENS
136:news
73:and
1844:Jay
1693:".
398:MFH
371:the
363:the
300:Mid
176:TWL
1863::
1849:💬
1809:)
1786:)
1763:)
1739:)
1731:.
1701:)
1688:↦
1677:)
1621:→
1584:→
1573:−
1443:)
1311:)
1299:.)
1237:)
1116:)
1063:
996:×
953:)
864:)
808:)
786:×
728:)
714:U(
699:U(
679:)
664:)
656:--
627:)
583:)
554:we
541:)
483:)
445:)
417:)
339::
156:)
54:;
1805:(
1799:R
1795:F
1782:(
1776:F
1770:F
1759:(
1735:(
1729:R
1725:R
1697:(
1690:R
1686:R
1673:(
1655:a
1650:G
1638:R
1624:R
1618:]
1615:t
1612:[
1608:Z
1587:R
1581:]
1576:1
1569:t
1565:,
1562:t
1559:[
1555:Z
1532:m
1527:G
1503:m
1498:G
1475:)
1472:R
1469:(
1464:m
1459:G
1439:(
1421:a
1416:G
1390:m
1385:G
1361:m
1356:G
1332:m
1327:G
1307:(
1285:a
1280:G
1256:m
1251:G
1233:(
1217:)
1213:C
1209:(
1204:m
1199:G
1171:a
1166:G
1138:m
1133:G
1112:(
1095:Q
1074:)
1070:Z
1066:(
1036:)
1033:R
1030:(
1025:m
1020:G
992:R
988:=
985:)
982:R
979:(
974:m
969:G
949:(
933:)
930:R
927:(
922:m
917:G
905:R
901:R
885:m
880:G
860:(
840:)
837:R
834:(
829:m
824:G
804:(
782:R
761:)
758:R
755:(
750:m
745:G
724:(
718:)
716:C
709:C
703:)
701:R
694:R
675:(
660:(
623:(
579:(
537:(
479:(
441:(
413:(
400::
356:a
312:.
211::
172:·
166:·
158:·
151:·
145:·
139:·
133:·
128:(
58:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
