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1016:
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587:
1021:
697:
1240:
748:
287:
255:
223:
191:
159:
445:
345:
128:
1162:{\displaystyle {\begin{aligned}s\cdot 0&=s\cdot (0+0)=s\cdot 0+s\cdot 0\\\Rightarrow s\cdot 0&=s\cdot 0-s\cdot 0\\\Rightarrow s\cdot 0&=0.\end{aligned}}}
898:
475:
is a group under the operation of addition and thus these also have a unique additive identity 0. This is defined to be different from the
482:
if the ring (or field) has more than one element. If the additive identity and the multiplicative identity are the same, then the ring is
596:
551:
1343:
1181:
be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, i.e. 0 = 1. Let
804:
672:
1195:
1381:
721:
59:, but additive identities occur in other mathematical structures where addition is defined, such as in
1376:
267:
235:
203:
171:
142:
1288:
1172:
476:
366:
32:
403:
303:
89:
664:
77:
56:
1356:
1386:
988:
483:
40:
1391:
972:{\displaystyle {\color {green}0'}={\color {green}0'}+0=0'+{\color {red}0}={\color {red}0}.}
515:
8:
1396:
472:
457:
362:
60:
713:
468:
64:
1308:
1339:
992:
702:
28:
1283:
1278:
461:
196:
1257:
709:
260:
135:
1173:
The additive and multiplicative identities are different in a non-trivial ring
1370:
753:
657:
228:
20:
1360:
1273:
479:
81:
52:
464:
of the group, is often denoted 0, and is unique (see below for proof).
542:
matrix whose entries consist entirely of the identity element 0 in
370:
36:
982:
164:
645:{\displaystyle 0={\begin{bmatrix}0&0\\0&0\end{bmatrix}}}
292:
1264:
is non-trivial then 0 is not equal to 1, is therefore shown.
765:
51:. One of the most familiar additive identities is the number
991:
over addition, the additive identity is a multiplicative
374:
611:
1198:
1019:
901:
807:
724:
675:
599:
554:
546:. For example, in the 2×2 matrices over the integers
406:
306:
270:
238:
206:
174:
145:
92:
582:{\displaystyle \operatorname {M} _{2}(\mathbb {Z} )}
522:, the additive identity is the zero matrix, denoted
1234:
1161:
971:
881:
742:
691:
644:
581:
439:
339:
291:the additive identity is 0. This says that for a
281:
249:
217:
185:
153:
122:
987:In a system with a multiplication operation that
1368:
983:The additive identity annihilates ring elements
882:{\displaystyle 0+g=g=g+0,\qquad 0'+g=g=g+0'.}
790:both denote additive identities, so for any
692:{\displaystyle \mathbb {R} \to \mathbb {R} }
705:every number to 0 is the additive identity.
766:The additive identity is unique in a group
1357:Uniqueness of additive identity in a ring
727:
685:
677:
572:
272:
240:
208:
176:
147:
1235:{\displaystyle r=r\times 1=r\times 0=0}
1369:
70:
1306:
16:Value that makes no change when added
892:It then follows from the above that
352:
76:The additive identity familiar from
1334:David S. Dummit, Richard M. Foote,
918:
903:
450:
13:
556:
14:
1408:
1350:
960:
950:
743:{\displaystyle \mathbb {R} ^{n},}
43:which, when added to any element
298:belonging to any of these sets,
1328:
838:
460:, the additive identity is the
1300:
1133:
1089:
1058:
1046:
681:
576:
568:
1:
1294:
760:
660:, 0 is the additive identity.
282:{\displaystyle \mathbb {C} ,}
250:{\displaystyle \mathbb {R} ,}
218:{\displaystyle \mathbb {Q} ,}
186:{\displaystyle \mathbb {Z} ,}
154:{\displaystyle \mathbb {N} }
7:
1267:
377:. An additive identity for
10:
1413:
440:{\displaystyle e+n=n=n+e.}
389:such that for any element
340:{\displaystyle n+0=n=0+n.}
123:{\displaystyle 5+0=5=0+5.}
31:that is equipped with the
1338:, Wiley (3rd ed.): 2003,
756:is the additive identity.
591:the additive identity is
365:that is closed under the
1010:. This follows because:
163:(if 0 is included), the
1289:Multiplicative identity
995:, meaning that for any
477:multiplicative identity
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1163:
973:
883:
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693:
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78:elementary mathematics
57:elementary mathematics
1313:mathworld.wolfram.com
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143:
90:
1309:"Additive Identity"
1307:Weisstein, Eric W.
778:be a group and let
385:, is an element in
71:Elementary examples
47:in the set, yields
1382:Elementary algebra
1232:
1185:be any element of
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1249:is trivial, i.e.
993:absorbing element
353:Formal definition
80:is zero, denoted
25:additive identity
1404:
1377:Abstract algebra
1336:Abstract Algebra
1323:
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1284:Identity element
1279:Additive inverse
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462:identity element
451:Further examples
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50:
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1401:
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731:
726:
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723:
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719:
717:
701:, the function
684:
676:
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671:
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663:In the ring of
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512:
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486:(proved below).
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261:complex numbers
239:
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207:
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175:
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146:
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136:natural numbers
91:
88:
87:
84:. For example,
73:
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17:
12:
11:
5:
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1351:External links
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1258:contrapositive
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752:the origin or
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710:additive group
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1344:0-471-43334-9
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1253:
1245:proving that
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1226:
1223:
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1199:
1192:
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1007:
994:
990:
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961:
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951:
946:
942:
939:
935:
932:
929:
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920:
914:
908:
905:
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893:
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869:
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534:, and is the
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83:
79:
75:
74:
68:
66:
62:
58:
54:
42:
38:
34:
30:
26:
22:
1387:Group theory
1335:
1329:Bibliography
1316:. Retrieved
1312:
1302:
1251:
1244:
1176:
1005:
986:
891:
773:
769:
530:
524:
518:over a ring
502:
497:
493:
489:In the ring
356:
229:real numbers
24:
18:
1392:Ring theory
989:distributes
754:zero vector
658:quaternions
21:mathematics
1397:0 (number)
1371:Categories
1361:PlanetMath
1318:2020-09-07
1295:References
1274:0 (number)
1260:, that if
761:Properties
381:, denoted
373:, denoted
1221:×
1209:×
1140:⋅
1134:⇒
1124:⋅
1118:−
1112:⋅
1096:⋅
1090:⇒
1080:⋅
1068:⋅
1044:⋅
1028:⋅
682:→
665:functions
566:
367:operation
33:operation
1268:See also
943:′
924:′
909:′
873:′
844:′
516:matrices
371:addition
259:and the
165:integers
37:addition
1189:. Then
1008:· 0 = 0
750:
718:
714:vectors
708:In the
703:mapping
699:
669:
656:In the
589:
548:
484:trivial
289:
264:
257:
232:
225:
200:
193:
168:
161:
139:
134:In the
41:element
1342:
1254:= {0}.
293:number
61:groups
39:is an
23:, the
667:from
473:field
458:group
456:In a
363:group
361:be a
65:rings
55:from
27:of a
1340:ISBN
1256:The
1177:Let
782:and
776:, +)
770:Let
538:-by-
511:-by-
469:ring
357:Let
227:the
195:the
63:and
1359:at
999:in
794:in
786:in
716:in
712:of
528:or
507:of
471:or
393:in
369:of
35:of
29:set
19:In
1373::
1311:.
1153:0.
1003:,
798:,
784:0'
496:×
467:A
397:,
118:5.
67:.
1363:.
1346:.
1321:.
1262:R
1252:R
1247:R
1230:0
1227:=
1224:0
1218:r
1215:=
1212:1
1206:r
1203:=
1200:r
1187:R
1183:r
1179:R
1150:=
1143:0
1137:s
1127:0
1121:s
1115:0
1109:s
1106:=
1099:0
1093:s
1083:0
1077:s
1074:+
1071:0
1065:s
1062:=
1059:)
1056:0
1053:+
1050:0
1047:(
1041:s
1038:=
1031:0
1025:s
1006:s
1001:S
997:s
967:.
962:0
957:=
952:0
947:+
940:0
936:=
933:0
930:+
921:0
915:=
906:0
877:.
870:0
866:+
863:g
860:=
857:g
854:=
851:g
848:+
841:0
836:,
833:0
830:+
827:g
824:=
821:g
818:=
815:g
812:+
809:0
796:G
792:g
788:G
780:0
774:G
772:(
738:,
733:n
728:R
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678:R
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632:0
627:0
620:0
615:0
609:[
604:=
601:0
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573:Z
569:(
561:2
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544:R
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498:n
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480:1
435:.
432:e
429:+
426:n
423:=
420:n
417:=
414:n
411:+
408:e
395:N
391:n
387:N
383:e
379:N
375:+
359:N
335:.
332:n
329:+
326:0
323:=
320:n
317:=
314:0
311:+
308:n
296:n
277:,
273:C
245:,
241:R
213:,
209:Q
181:,
177:Z
148:N
115:+
112:0
109:=
106:5
103:=
100:0
97:+
94:5
82:0
53:0
49:x
45:x
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