32:
1603:
1459:
196:
1454:
1392:
2508:
1705:, has nontrivial zero divisors: there are pairs of functions which are not identically equal to zero yet whose product is the zero function. In fact, it is not hard to construct, for any
1056:
2409:
1322:
1284:
2554:
1753:
1695:
2162:
913:
473:
2801:
1112:
1008:
956:
2772:
2224:
1179:
2823:
1791:
329:
304:
279:
254:
1944:
2637:
2297:
1817:
2034:
1999:
1141:
649:
1598:{\displaystyle MN={\begin{pmatrix}1&-1\\0&0\end{pmatrix}}{\begin{pmatrix}0&1\\0&1\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}=0,}
701:
675:
2057:
2720:
2697:
2677:
2657:
2598:
2578:
2429:
2357:
2337:
2317:
2267:
2247:
2097:
2077:
1964:
1897:
1877:
1857:
1643:
1623:
1239:
1219:
1199:
1079:
978:
843:
817:
761:
741:
721:
620:
600:
580:
560:
540:
520:
500:
425:
405:
385:
365:
1335:
42:
136:
1397:
792:
is an integral domain; in fact, any subring of a field is an integral domain (as long as it contains 1). Similarly, any subring of a
2911:
100:
72:
2834:
79:
57:
2103:
86:
2944:
868:, the zero-product property holds. This ring is not an integral domain, because the multiplication is not commutative.
2434:
68:
2939:
1013:
789:
2362:
1292:
1244:
2949:
1649:
228:
2513:
1712:
1654:
2109:
874:
434:
2777:
1088:
984:
932:
2954:
2843:
2725:
2167:
1325:
773:
336:
2959:
1146:
93:
2806:
1758:
312:
287:
262:
237:
2722:
is not an integral domain, then the conclusion need not hold. For example, the cubic polynomial
959:
824:
428:
332:
1902:
2603:
2907:
2886:
1827:
850:
2276:
1796:
2270:
2004:
1969:
1082:
777:
2839:
1823:
1117:
854:
785:
625:
407:
must have both additive structure and multiplicative structure. Usually one assumes that
257:
1966:
to come from any integral domain.) By the zero-product property, it follows that either
680:
654:
2039:
1822:
The same is true even if we consider only continuous functions, or only even infinitely
2705:
2682:
2662:
2642:
2583:
2563:
2414:
2342:
2322:
2302:
2252:
2232:
2082:
2062:
1949:
1882:
1862:
1842:
1628:
1608:
1224:
1204:
1184:
1064:
963:
828:
802:
746:
726:
706:
605:
585:
565:
545:
525:
505:
485:
410:
390:
370:
350:
307:
2933:
1698:
224:
20:
796:
is a domain. Thus, the zero-product property holds for any subring of a skew field.
49:
2853:
820:
781:
215:
130:
2848:
1702:
865:
282:
31:
1010:
does not satisfy the zero product property: 2 and 3 are nonzero elements, yet
861:
793:
387:
have the zero-product property? In order for this question to have meaning,
2924:
431:, though it could be something else, e.g. the set of nonnegative integers
916:
476:
475:
with ordinary addition and multiplication, which is only a (commutative)
1329:
232:
122:
2872:
The other being a⋅0 = 0⋅a = 0. Mustafa A. Munem and David J. Foulis,
191:{\displaystyle {\text{if }}ab=0,{\text{ then }}a=0{\text{ or }}b=0.}
1449:{\displaystyle N={\begin{pmatrix}0&1\\0&1\end{pmatrix}},}
1387:{\displaystyle M={\begin{pmatrix}1&-1\\0&0\end{pmatrix}}}
2106:
to find the roots of a polynomial. For example, the polynomial
25:
772:
A ring in which the zero-product property holds is called a
331:— satisfy the zero-product property. In general, a
1114:
does not satisfy the zero-product property. Namely, if
1879:
are univariate polynomials with real coefficients, and
1332:
entries does not satisfy the zero-product property: if
846:
has the zero-product property (in fact, it is a field).
53:
1555:
1516:
1477:
1412:
1350:
335:
which satisfies the zero-product property is called a
2809:
2780:
2728:
2708:
2685:
2665:
2645:
2606:
2586:
2566:
2516:
2437:
2417:
2365:
2345:
2325:
2305:
2279:
2255:
2235:
2170:
2112:
2085:
2065:
2042:
2007:
1972:
1952:
1905:
1885:
1865:
1845:
1799:
1761:
1715:
1657:
1631:
1611:
1462:
1400:
1338:
1295:
1247:
1227:
1207:
1187:
1149:
1120:
1091:
1067:
1016:
987:
966:
935:
877:
831:
805:
749:
729:
709:
683:
657:
628:
608:
588:
568:
548:
528:
508:
488:
437:
413:
393:
373:
353:
315:
290:
265:
240:
139:
2817:
2795:
2766:
2714:
2691:
2671:
2651:
2631:
2592:
2572:
2548:
2502:
2423:
2403:
2351:
2331:
2311:
2291:
2261:
2241:
2218:
2156:
2091:
2071:
2051:
2028:
1993:
1958:
1938:
1891:
1871:
1851:
1811:
1785:
1747:
1689:
1637:
1617:
1597:
1448:
1386:
1316:
1278:
1233:
1213:
1193:
1173:
1135:
1106:
1073:
1050:
1002:
972:
950:
907:
857:because they are a subring of the complex numbers.
837:
811:
755:
735:
715:
695:
669:
643:
614:
594:
574:
554:
534:
514:
494:
467:
419:
399:
379:
359:
323:
298:
273:
248:
190:
2889:) and a notion of products, i.e., multiplication.
2510:. By the zero-product property, it follows that
919:), but it does satisfy the zero-product property.
1946:. (Actually, we may allow the coefficients and
2226:; hence, its roots are precisely 3, 1, and −2.
1755:, none of which is identically zero, such that
367:is an algebraic structure. We might ask, does
2503:{\displaystyle f(x)=(x-r_{1})\cdots (x-r_{d})}
2411:. It follows (but we do not prove here) that
16:The product of two nonzero elements is nonzero
562:also satisfies the zero product property: if
8:
902:
878:
502:satisfies the zero-product property, and if
462:
438:
58:introducing citations to additional sources
1835:Application to finding roots of polynomials
1051:{\displaystyle 2\cdot 3\equiv 0{\pmod {6}}}
2876:(New York: Worth Publishers, 1982), p. 4.
2874:Algebra and Trigonometry with Applications
2811:
2810:
2808:
2787:
2783:
2782:
2779:
2749:
2733:
2727:
2707:
2684:
2664:
2644:
2620:
2605:
2585:
2565:
2540:
2521:
2515:
2491:
2466:
2436:
2416:
2389:
2370:
2364:
2344:
2324:
2304:
2278:
2254:
2234:
2169:
2133:
2117:
2111:
2084:
2064:
2041:
2006:
1971:
1951:
1904:
1884:
1864:
1844:
1798:
1777:
1772:
1766:
1760:
1739:
1720:
1714:
1683:
1682:
1656:
1630:
1610:
1550:
1511:
1472:
1461:
1407:
1399:
1345:
1337:
1302:
1298:
1297:
1294:
1260:
1246:
1226:
1206:
1186:
1148:
1119:
1098:
1094:
1093:
1090:
1066:
1032:
1015:
994:
990:
989:
986:
965:
942:
938:
937:
934:
876:
830:
804:
748:
728:
708:
682:
656:
627:
607:
587:
567:
547:
527:
507:
487:
436:
412:
392:
372:
352:
317:
316:
314:
292:
291:
289:
267:
266:
264:
242:
241:
239:
174:
160:
140:
138:
2404:{\displaystyle r_{1},\ldots ,r_{d}\in R}
1317:{\displaystyle \mathbb {Z} ^{2\times 2}}
48:Relevant discussion may be found on the
2865:
2902:David S. Dummit and Richard M. Foote,
743:can also be considered as elements of
1279:{\displaystyle qm\equiv 0{\pmod {n}}}
19:For the product of zero factors, see
7:
2885:There must be a notion of zero (the
2803:(though it has only three roots in
2549:{\displaystyle r_{1},\ldots ,r_{d}}
1748:{\displaystyle f_{1},\ldots ,f_{n}}
1268:
1040:
200:This property is also known as the
1690:{\displaystyle f:\to \mathbb {R} }
14:
2157:{\displaystyle x^{3}-2x^{2}-5x+6}
908:{\displaystyle \{0,1,2,\ldots \}}
468:{\displaystyle \{0,1,2,\ldots \}}
2925:PlanetMath: Zero rule of product
2796:{\displaystyle \mathbb {Z} _{6}}
2273:univariate polynomial of degree
2036:. In other words, the roots of
1107:{\displaystyle \mathbb {Z} _{n}}
1003:{\displaystyle \mathbb {Z} _{6}}
951:{\displaystyle \mathbb {Z} _{n}}
871:The set of nonnegative integers
41:relies largely or entirely on a
30:
2767:{\displaystyle x^{3}+3x^{2}+2x}
2219:{\displaystyle (x-3)(x-1)(x+2)}
1830:have the zero-product property.
1261:
1033:
915:is not a ring (being instead a
210:multiplication property of zero
129:states that the product of two
2835:Fundamental theorem of algebra
2626:
2607:
2497:
2478:
2472:
2453:
2447:
2441:
2213:
2201:
2198:
2186:
2183:
2171:
2017:
2011:
1982:
1976:
1927:
1921:
1915:
1909:
1679:
1676:
1664:
1272:
1262:
1044:
1034:
1:
1793:is identically zero whenever
1174:{\displaystyle 0<q,m<n}
2818:{\displaystyle \mathbb {Z} }
1786:{\displaystyle f_{i}\,f_{j}}
324:{\displaystyle \mathbb {C} }
299:{\displaystyle \mathbb {R} }
274:{\displaystyle \mathbb {Q} }
249:{\displaystyle \mathbb {Z} }
133:is nonzero. In other words,
2079:together with the roots of
2059:are precisely the roots of
1899:is a real number such that
214:nonexistence of nontrivial
2976:
2249:is an integral domain and
1939:{\displaystyle P(x)Q(x)=0}
18:
2632:{\displaystyle (x-r_{i})}
2906:(3d ed.), Wiley, 2003,
2292:{\displaystyle d\geq 1}
1812:{\displaystyle i\neq j}
782:multiplicative identity
69:"Zero-product property"
2819:
2797:
2768:
2716:
2693:
2673:
2653:
2633:
2594:
2574:
2550:
2504:
2425:
2405:
2353:
2333:
2313:
2293:
2263:
2243:
2220:
2158:
2093:
2073:
2053:
2030:
2029:{\displaystyle Q(x)=0}
1995:
1994:{\displaystyle P(x)=0}
1960:
1940:
1893:
1873:
1853:
1813:
1787:
1749:
1691:
1639:
1619:
1599:
1450:
1388:
1318:
1280:
1235:
1215:
1195:
1175:
1137:
1108:
1075:
1052:
1004:
974:
952:
909:
839:
813:
757:
737:
717:
697:
671:
645:
616:
596:
576:
556:
536:
516:
496:
469:
421:
401:
381:
361:
325:
300:
275:
250:
229:elementary mathematics
221:zero-factor properties
192:
2820:
2798:
2769:
2717:
2694:
2674:
2654:
2634:
2595:
2575:
2551:
2505:
2426:
2406:
2354:
2334:
2319:. Suppose also that
2314:
2299:with coefficients in
2294:
2264:
2244:
2221:
2159:
2094:
2074:
2054:
2031:
1996:
1961:
1941:
1894:
1874:
1854:
1826:. On the other hand,
1814:
1788:
1750:
1692:
1640:
1620:
1600:
1451:
1389:
1319:
1281:
1236:
1216:
1196:
1176:
1138:
1109:
1076:
1053:
1005:
975:
953:
910:
840:
814:
784:element is called an
758:
738:
718:
698:
672:
646:
617:
597:
577:
557:
537:
517:
497:
470:
422:
402:
382:
362:
326:
301:
276:
251:
193:
127:zero-product property
2807:
2778:
2726:
2706:
2683:
2663:
2643:
2604:
2584:
2564:
2514:
2435:
2415:
2363:
2343:
2323:
2303:
2277:
2253:
2233:
2229:In general, suppose
2168:
2110:
2083:
2063:
2040:
2005:
1970:
1950:
1903:
1883:
1863:
1843:
1797:
1759:
1713:
1655:
1629:
1609:
1460:
1398:
1336:
1293:
1245:
1225:
1205:
1185:
1147:
1136:{\displaystyle n=qm}
1118:
1089:
1065:
1014:
985:
964:
933:
875:
829:
803:
747:
727:
707:
681:
655:
644:{\displaystyle ab=0}
626:
606:
586:
566:
546:
526:
506:
486:
435:
411:
391:
371:
351:
313:
288:
263:
238:
219:, or one of the two
202:rule of zero product
137:
54:improve this article
1221:are nonzero modulo
958:denote the ring of
862:strictly skew field
823:, then the ring of
696:{\displaystyle b=0}
670:{\displaystyle a=0}
2945:Elementary algebra
2815:
2793:
2764:
2712:
2689:
2669:
2659:. In particular,
2649:
2629:
2600:must be a root of
2590:
2570:
2546:
2500:
2421:
2401:
2349:
2329:
2309:
2289:
2259:
2239:
2216:
2154:
2102:Thus, one can use
2089:
2069:
2052:{\displaystyle PQ}
2049:
2026:
1991:
1956:
1936:
1889:
1869:
1849:
1828:analytic functions
1809:
1783:
1745:
1687:
1635:
1615:
1595:
1580:
1541:
1505:
1446:
1437:
1384:
1378:
1314:
1276:
1231:
1211:
1191:
1171:
1133:
1104:
1071:
1048:
1000:
970:
948:
905:
835:
809:
753:
733:
713:
693:
667:
641:
612:
592:
572:
552:
532:
512:
492:
465:
417:
397:
377:
357:
321:
296:
271:
246:
188:
2887:additive identity
2774:has six roots in
2715:{\displaystyle R}
2692:{\displaystyle d}
2672:{\displaystyle f}
2652:{\displaystyle i}
2593:{\displaystyle f}
2573:{\displaystyle f}
2424:{\displaystyle f}
2352:{\displaystyle d}
2332:{\displaystyle f}
2312:{\displaystyle R}
2262:{\displaystyle f}
2242:{\displaystyle R}
2092:{\displaystyle Q}
2072:{\displaystyle P}
1959:{\displaystyle x}
1892:{\displaystyle x}
1872:{\displaystyle Q}
1852:{\displaystyle P}
1638:{\displaystyle N}
1618:{\displaystyle M}
1234:{\displaystyle n}
1214:{\displaystyle q}
1194:{\displaystyle m}
1074:{\displaystyle n}
973:{\displaystyle n}
851:Gaussian integers
838:{\displaystyle p}
812:{\displaystyle p}
756:{\displaystyle A}
736:{\displaystyle b}
716:{\displaystyle a}
615:{\displaystyle B}
595:{\displaystyle b}
575:{\displaystyle a}
555:{\displaystyle B}
535:{\displaystyle A}
515:{\displaystyle B}
495:{\displaystyle A}
420:{\displaystyle A}
400:{\displaystyle A}
380:{\displaystyle A}
360:{\displaystyle A}
343:Algebraic context
177:
163:
143:
119:
118:
104:
2967:
2940:Abstract algebra
2904:Abstract Algebra
2890:
2883:
2877:
2870:
2824:
2822:
2821:
2816:
2814:
2802:
2800:
2799:
2794:
2792:
2791:
2786:
2773:
2771:
2770:
2765:
2754:
2753:
2738:
2737:
2721:
2719:
2718:
2713:
2699:distinct roots.
2698:
2696:
2695:
2690:
2678:
2676:
2675:
2670:
2658:
2656:
2655:
2650:
2638:
2636:
2635:
2630:
2625:
2624:
2599:
2597:
2596:
2591:
2579:
2577:
2576:
2571:
2555:
2553:
2552:
2547:
2545:
2544:
2526:
2525:
2509:
2507:
2506:
2501:
2496:
2495:
2471:
2470:
2430:
2428:
2427:
2422:
2410:
2408:
2407:
2402:
2394:
2393:
2375:
2374:
2358:
2356:
2355:
2350:
2338:
2336:
2335:
2330:
2318:
2316:
2315:
2310:
2298:
2296:
2295:
2290:
2268:
2266:
2265:
2260:
2248:
2246:
2245:
2240:
2225:
2223:
2222:
2217:
2163:
2161:
2160:
2155:
2138:
2137:
2122:
2121:
2098:
2096:
2095:
2090:
2078:
2076:
2075:
2070:
2058:
2056:
2055:
2050:
2035:
2033:
2032:
2027:
2000:
1998:
1997:
1992:
1965:
1963:
1962:
1957:
1945:
1943:
1942:
1937:
1898:
1896:
1895:
1890:
1878:
1876:
1875:
1870:
1858:
1856:
1855:
1850:
1824:smooth functions
1818:
1816:
1815:
1810:
1792:
1790:
1789:
1784:
1782:
1781:
1771:
1770:
1754:
1752:
1751:
1746:
1744:
1743:
1725:
1724:
1696:
1694:
1693:
1688:
1686:
1648:The ring of all
1644:
1642:
1641:
1636:
1624:
1622:
1621:
1616:
1604:
1602:
1601:
1596:
1585:
1584:
1546:
1545:
1510:
1509:
1455:
1453:
1452:
1447:
1442:
1441:
1393:
1391:
1390:
1385:
1383:
1382:
1323:
1321:
1320:
1315:
1313:
1312:
1301:
1285:
1283:
1282:
1277:
1275:
1240:
1238:
1237:
1232:
1220:
1218:
1217:
1212:
1200:
1198:
1197:
1192:
1180:
1178:
1177:
1172:
1142:
1140:
1139:
1134:
1113:
1111:
1110:
1105:
1103:
1102:
1097:
1083:composite number
1080:
1078:
1077:
1072:
1057:
1055:
1054:
1049:
1047:
1009:
1007:
1006:
1001:
999:
998:
993:
979:
977:
976:
971:
960:integers modulo
957:
955:
954:
949:
947:
946:
941:
914:
912:
911:
906:
844:
842:
841:
836:
825:integers modulo
818:
816:
815:
810:
762:
760:
759:
754:
742:
740:
739:
734:
722:
720:
719:
714:
702:
700:
699:
694:
676:
674:
673:
668:
650:
648:
647:
642:
621:
619:
618:
613:
602:are elements of
601:
599:
598:
593:
581:
579:
578:
573:
561:
559:
558:
553:
541:
539:
538:
533:
521:
519:
518:
513:
501:
499:
498:
493:
474:
472:
471:
466:
426:
424:
423:
418:
406:
404:
403:
398:
386:
384:
383:
378:
366:
364:
363:
358:
330:
328:
327:
322:
320:
305:
303:
302:
297:
295:
280:
278:
277:
272:
270:
258:rational numbers
255:
253:
252:
247:
245:
197:
195:
194:
189:
178:
175:
164:
162: then
161:
144:
141:
131:nonzero elements
114:
111:
105:
103:
62:
34:
26:
2975:
2974:
2970:
2969:
2968:
2966:
2965:
2964:
2930:
2929:
2921:
2899:
2894:
2893:
2884:
2880:
2871:
2867:
2862:
2840:Integral domain
2831:
2805:
2804:
2781:
2776:
2775:
2745:
2729:
2724:
2723:
2704:
2703:
2681:
2680:
2661:
2660:
2641:
2640:
2616:
2602:
2601:
2582:
2581:
2562:
2561:
2536:
2517:
2512:
2511:
2487:
2462:
2433:
2432:
2413:
2412:
2385:
2366:
2361:
2360:
2359:distinct roots
2341:
2340:
2321:
2320:
2301:
2300:
2275:
2274:
2251:
2250:
2231:
2230:
2166:
2165:
2129:
2113:
2108:
2107:
2081:
2080:
2061:
2060:
2038:
2037:
2003:
2002:
1968:
1967:
1948:
1947:
1901:
1900:
1881:
1880:
1861:
1860:
1841:
1840:
1837:
1795:
1794:
1773:
1762:
1757:
1756:
1735:
1716:
1711:
1710:
1709:≥ 2, functions
1653:
1652:
1627:
1626:
1607:
1606:
1579:
1578:
1573:
1567:
1566:
1561:
1551:
1540:
1539:
1534:
1528:
1527:
1522:
1512:
1504:
1503:
1498:
1492:
1491:
1483:
1473:
1458:
1457:
1436:
1435:
1430:
1424:
1423:
1418:
1408:
1396:
1395:
1377:
1376:
1371:
1365:
1364:
1356:
1346:
1334:
1333:
1296:
1291:
1290:
1243:
1242:
1223:
1222:
1203:
1202:
1183:
1182:
1145:
1144:
1116:
1115:
1092:
1087:
1086:
1063:
1062:
1061:In general, if
1012:
1011:
988:
983:
982:
962:
961:
936:
931:
930:
926:
873:
872:
855:integral domain
827:
826:
801:
800:
786:integral domain
769:
745:
744:
725:
724:
705:
704:
679:
678:
653:
652:
624:
623:
604:
603:
584:
583:
564:
563:
544:
543:
524:
523:
522:is a subset of
504:
503:
484:
483:
433:
432:
409:
408:
389:
388:
369:
368:
349:
348:
345:
311:
310:
308:complex numbers
286:
285:
261:
260:
236:
235:
206:null factor law
135:
134:
115:
109:
106:
63:
61:
47:
35:
24:
17:
12:
11:
5:
2973:
2971:
2963:
2962:
2957:
2952:
2947:
2942:
2932:
2931:
2928:
2927:
2920:
2919:External links
2917:
2916:
2915:
2898:
2895:
2892:
2891:
2878:
2864:
2863:
2861:
2858:
2857:
2856:
2851:
2846:
2837:
2830:
2827:
2813:
2790:
2785:
2763:
2760:
2757:
2752:
2748:
2744:
2741:
2736:
2732:
2711:
2688:
2668:
2648:
2628:
2623:
2619:
2615:
2612:
2609:
2589:
2580:: any root of
2569:
2543:
2539:
2535:
2532:
2529:
2524:
2520:
2499:
2494:
2490:
2486:
2483:
2480:
2477:
2474:
2469:
2465:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2431:factorizes as
2420:
2400:
2397:
2392:
2388:
2384:
2381:
2378:
2373:
2369:
2348:
2328:
2308:
2288:
2285:
2282:
2258:
2238:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2164:factorizes as
2153:
2150:
2147:
2144:
2141:
2136:
2132:
2128:
2125:
2120:
2116:
2088:
2068:
2048:
2045:
2025:
2022:
2019:
2016:
2013:
2010:
1990:
1987:
1984:
1981:
1978:
1975:
1955:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1888:
1868:
1848:
1836:
1833:
1832:
1831:
1820:
1808:
1805:
1802:
1780:
1776:
1769:
1765:
1742:
1738:
1734:
1731:
1728:
1723:
1719:
1685:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1646:
1634:
1614:
1594:
1591:
1588:
1583:
1577:
1574:
1572:
1569:
1568:
1565:
1562:
1560:
1557:
1556:
1554:
1549:
1544:
1538:
1535:
1533:
1530:
1529:
1526:
1523:
1521:
1518:
1517:
1515:
1508:
1502:
1499:
1497:
1494:
1493:
1490:
1487:
1484:
1482:
1479:
1478:
1476:
1471:
1468:
1465:
1445:
1440:
1434:
1431:
1429:
1426:
1425:
1422:
1419:
1417:
1414:
1413:
1411:
1406:
1403:
1381:
1375:
1372:
1370:
1367:
1366:
1363:
1360:
1357:
1355:
1352:
1351:
1349:
1344:
1341:
1311:
1308:
1305:
1300:
1287:
1274:
1271:
1267:
1264:
1259:
1256:
1253:
1250:
1230:
1210:
1190:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1132:
1129:
1126:
1123:
1101:
1096:
1070:
1059:
1046:
1043:
1039:
1036:
1031:
1028:
1025:
1022:
1019:
997:
992:
969:
945:
940:
925:
922:
921:
920:
904:
901:
898:
895:
892:
889:
886:
883:
880:
869:
858:
847:
834:
808:
797:
780:domain with a
768:
765:
752:
732:
712:
692:
689:
686:
666:
663:
660:
651:, then either
640:
637:
634:
631:
611:
591:
571:
551:
531:
511:
491:
464:
461:
458:
455:
452:
449:
446:
443:
440:
416:
396:
376:
356:
344:
341:
319:
294:
269:
244:
225:number systems
223:. All of the
187:
184:
181:
176: or
173:
170:
167:
159:
156:
153:
150:
147:
117:
116:
52:. Please help
38:
36:
29:
15:
13:
10:
9:
6:
4:
3:
2:
2972:
2961:
2958:
2956:
2953:
2951:
2950:Real analysis
2948:
2946:
2943:
2941:
2938:
2937:
2935:
2926:
2923:
2922:
2918:
2913:
2912:0-471-43334-9
2909:
2905:
2901:
2900:
2896:
2888:
2882:
2879:
2875:
2869:
2866:
2859:
2855:
2852:
2850:
2847:
2845:
2841:
2838:
2836:
2833:
2832:
2828:
2826:
2788:
2761:
2758:
2755:
2750:
2746:
2742:
2739:
2734:
2730:
2709:
2700:
2686:
2666:
2646:
2621:
2617:
2613:
2610:
2587:
2567:
2559:
2541:
2537:
2533:
2530:
2527:
2522:
2518:
2492:
2488:
2484:
2481:
2475:
2467:
2463:
2459:
2456:
2450:
2444:
2438:
2418:
2398:
2395:
2390:
2386:
2382:
2379:
2376:
2371:
2367:
2346:
2326:
2306:
2286:
2283:
2280:
2272:
2256:
2236:
2227:
2210:
2207:
2204:
2195:
2192:
2189:
2180:
2177:
2174:
2151:
2148:
2145:
2142:
2139:
2134:
2130:
2126:
2123:
2118:
2114:
2105:
2104:factorization
2100:
2086:
2066:
2046:
2043:
2023:
2020:
2014:
2008:
1988:
1985:
1979:
1973:
1953:
1933:
1930:
1924:
1918:
1912:
1906:
1886:
1866:
1846:
1834:
1829:
1825:
1821:
1806:
1803:
1800:
1778:
1774:
1767:
1763:
1740:
1736:
1732:
1729:
1726:
1721:
1717:
1708:
1704:
1700:
1699:unit interval
1673:
1670:
1667:
1661:
1658:
1651:
1647:
1632:
1612:
1592:
1589:
1586:
1581:
1575:
1570:
1563:
1558:
1552:
1547:
1542:
1536:
1531:
1524:
1519:
1513:
1506:
1500:
1495:
1488:
1485:
1480:
1474:
1469:
1466:
1463:
1443:
1438:
1432:
1427:
1420:
1415:
1409:
1404:
1401:
1379:
1373:
1368:
1361:
1358:
1353:
1347:
1342:
1339:
1331:
1327:
1309:
1306:
1303:
1288:
1269:
1265:
1257:
1254:
1251:
1248:
1228:
1208:
1188:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1130:
1127:
1124:
1121:
1099:
1084:
1068:
1060:
1041:
1037:
1029:
1026:
1023:
1020:
1017:
995:
980:
967:
943:
928:
927:
923:
918:
899:
896:
893:
890:
887:
884:
881:
870:
867:
863:
859:
856:
852:
848:
845:
832:
822:
806:
798:
795:
791:
787:
783:
779:
775:
771:
770:
766:
764:
750:
730:
710:
690:
687:
684:
664:
661:
658:
638:
635:
632:
629:
609:
589:
569:
549:
529:
509:
489:
482:Note that if
480:
478:
459:
456:
453:
450:
447:
444:
441:
430:
414:
394:
374:
354:
342:
340:
338:
334:
309:
284:
259:
234:
230:
226:
222:
218:
217:
216:zero divisors
211:
207:
203:
198:
185:
182:
179:
171:
168:
165:
157:
154:
151:
148:
145:
132:
128:
124:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
51:
45:
44:
43:single source
39:This article
37:
33:
28:
27:
22:
21:empty product
2903:
2881:
2873:
2868:
2854:Zero divisor
2701:
2679:has at most
2557:
2228:
2101:
1838:
1706:
1703:real numbers
1605:yet neither
924:Non-examples
821:prime number
481:
346:
283:real numbers
231:— the
220:
213:
209:
205:
201:
199:
126:
120:
107:
97:
90:
83:
76:
64:
40:
2955:Ring theory
2849:Prime ideal
2702:If however
1697:, from the
866:quaternions
778:commutative
227:studied in
2960:0 (number)
2934:Categories
2897:References
794:skew field
622:such that
306:, and the
80:newspapers
2639:for some
2614:−
2560:roots of
2531:…
2485:−
2476:⋯
2460:−
2396:∈
2380:…
2284:≥
2193:−
2178:−
2140:−
2124:−
1804:≠
1730:…
1680:→
1650:functions
1486:−
1359:−
1307:×
1289:The ring
1255:≡
1027:≡
1021:⋅
900:…
460:…
50:talk page
2829:See also
2556:are the
1839:Suppose
1645:is zero.
1326:matrices
917:semiring
767:Examples
703:because
477:semiring
347:Suppose
233:integers
142:if
110:May 2022
1701:to the
1330:integer
1324:of 2×2
1181:, then
1085:, then
981:. Then
860:In the
853:are an
788:. Any
542:, then
123:algebra
94:scholar
2910:
2844:domain
1241:, yet
1143:where
774:domain
337:domain
281:, the
256:, the
212:, the
208:, the
204:, the
125:, the
96:
89:
82:
75:
67:
2860:Notes
2271:monic
2269:is a
1456:then
1328:with
1081:is a
819:is a
790:field
776:. A
427:is a
101:JSTOR
87:books
2908:ISBN
2842:and
2558:only
2339:has
1859:and
1625:nor
1394:and
1201:and
1166:<
1154:<
929:Let
849:The
723:and
582:and
429:ring
333:ring
73:news
2825:).
2001:or
1266:mod
1038:mod
864:of
799:If
677:or
121:In
56:by
2936::
2099:.
763:.
479:.
339:.
186:0.
2914:.
2812:Z
2789:6
2784:Z
2762:x
2759:2
2756:+
2751:2
2747:x
2743:3
2740:+
2735:3
2731:x
2710:R
2687:d
2667:f
2647:i
2627:)
2622:i
2618:r
2611:x
2608:(
2588:f
2568:f
2542:d
2538:r
2534:,
2528:,
2523:1
2519:r
2498:)
2493:d
2489:r
2482:x
2479:(
2473:)
2468:1
2464:r
2457:x
2454:(
2451:=
2448:)
2445:x
2442:(
2439:f
2419:f
2399:R
2391:d
2387:r
2383:,
2377:,
2372:1
2368:r
2347:d
2327:f
2307:R
2287:1
2281:d
2257:f
2237:R
2214:)
2211:2
2208:+
2205:x
2202:(
2199:)
2196:1
2190:x
2187:(
2184:)
2181:3
2175:x
2172:(
2152:6
2149:+
2146:x
2143:5
2135:2
2131:x
2127:2
2119:3
2115:x
2087:Q
2067:P
2047:Q
2044:P
2024:0
2021:=
2018:)
2015:x
2012:(
2009:Q
1989:0
1986:=
1983:)
1980:x
1977:(
1974:P
1954:x
1934:0
1931:=
1928:)
1925:x
1922:(
1919:Q
1916:)
1913:x
1910:(
1907:P
1887:x
1867:Q
1847:P
1819:.
1807:j
1801:i
1779:j
1775:f
1768:i
1764:f
1741:n
1737:f
1733:,
1727:,
1722:1
1718:f
1707:n
1684:R
1677:]
1674:1
1671:,
1668:0
1665:[
1662::
1659:f
1633:N
1613:M
1593:,
1590:0
1587:=
1582:)
1576:0
1571:0
1564:0
1559:0
1553:(
1548:=
1543:)
1537:1
1532:0
1525:1
1520:0
1514:(
1507:)
1501:0
1496:0
1489:1
1481:1
1475:(
1470:=
1467:N
1464:M
1444:,
1439:)
1433:1
1428:0
1421:1
1416:0
1410:(
1405:=
1402:N
1380:)
1374:0
1369:0
1362:1
1354:1
1348:(
1343:=
1340:M
1310:2
1304:2
1299:Z
1286:.
1273:)
1270:n
1263:(
1258:0
1252:m
1249:q
1229:n
1209:q
1189:m
1169:n
1163:m
1160:,
1157:q
1151:0
1131:m
1128:q
1125:=
1122:n
1100:n
1095:Z
1069:n
1058:.
1045:)
1042:6
1035:(
1030:0
1024:3
1018:2
996:6
991:Z
968:n
944:n
939:Z
903:}
897:,
894:2
891:,
888:1
885:,
882:0
879:{
833:p
807:p
751:A
731:b
711:a
691:0
688:=
685:b
665:0
662:=
659:a
639:0
636:=
633:b
630:a
610:B
590:b
570:a
550:B
530:A
510:B
490:A
463:}
457:,
454:2
451:,
448:1
445:,
442:0
439:{
415:A
395:A
375:A
355:A
318:C
293:R
268:Q
243:Z
183:=
180:b
172:0
169:=
166:a
158:,
155:0
152:=
149:b
146:a
112:)
108:(
98:·
91:·
84:·
77:·
60:.
46:.
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
