736:, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral domains are sometimes admitted. This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "
3340:
4762:
of the ring is zero, so that the intersection of all the ring's minimal primes is zero. The latter condition is that the ring have only one minimal prime. It follows that the unique minimal prime ideal of a reduced and irreducible ring is the zero ideal, so such rings are integral domains. The
1468:
3771:
6331:
5147:
3085:
3634:
3547:
2832:
5230:
5562:
4326:
3716:
3456:
4990:
5817:
1386:
1822:
3898:
3841:
4595:, and, in algebraic geometry, it implies the statement that the coordinate ring of the product of two affine algebraic varieties over an algebraically closed field is again an integral domain.
1669:
5336:
5283:
253:
2552:
2155:
2344:
2308:
545:
2484:
5436:
5376:
2383:
4377:
4208:
2580:
2047:
1736:
1952:
498:
461:
5480:
5054:
4257:
4718:
4161:
1915:
4690:
3677:
2014:
1890:
1564:
1507:
1379:
1333:
1300:
207:
5030:
3721:
2416:
5837:
5683:
5650:
2994:
2907:
1868:
5737:
5710:
4911:
5591:
3035:
2767:
2192:
5617:
2632:
2606:
2117:
1158:
3409:
5456:
5396:
3386:
3366:
2927:
2738:
2718:
2698:
2678:
2272:
2252:
2232:
2212:
1982:
1842:
650:
3335:{\displaystyle f(x)={\begin{cases}1-2x&x\in \left\\0&x\in \left\end{cases}}\qquad g(x)={\begin{cases}0&x\in \left\\2x-1&x\in \left\end{cases}}}
5063:
3552:
3465:
4648:
modulo an appropriate equivalence relation, equipped with the usual addition and multiplication operations. It is "the smallest field containing
2772:
1381:
provides an example of a non-Artinian infinite integral domain that is not a field, possessing infinite descending sequences of ideals such as:
5152:
4210:
the element 3 is irreducible (if it factored nontrivially, the factors would each have to have norm 3, but there are no norm 3 elements since
6160:
6127:
6105:
6055:
6032:
732:"Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a
5485:
4262:
107:
1151:
3682:
3422:
6405:(1958). "A General Theory of Algebraic Geometry Over Dedekind Domains, II: Separably Generated Extensions and Regular Local Rings".
6303:
6276:
6253:
6234:
6194:
6074:
6006:
4336:
643:
595:
1463:{\displaystyle \mathbb {Z} \supset 2\mathbb {Z} \supset \cdots \supset 2^{n}\mathbb {Z} \supset 2^{n+1}\mathbb {Z} \supset \cdots }
5750:
4920:
6407:
6313:
6217:
4763:
converse is clear: an integral domain has no nonzero nilpotent elements, and the zero ideal is the unique minimal prime ideal.
1748:
3846:
3776:
1348:
1144:
6485:
4592:
636:
2069:
1580:
1013:
799:
353:
4384:
6295:
3954:
3914:
3417:
823:
691:
113:
217:
5288:
5235:
2492:
2125:
4805:
3901:
3459:
2313:
2277:
787:
588:
512:
391:
341:
4849:
2421:
1104:
733:
400:
93:
5401:
5341:
2349:
1577:
The previous example can be further exploited by taking quotients from prime ideals. For example, the ring
4824:
4551:
3066:
1739:
805:
557:
408:
359:
140:
4855:
1221:
1210:
755:
699:
679:
4342:
4173:
3008:
in this quotient ring are nonzero elements whose product is 0. This argument shows, equivalently, that
2557:
2019:
1678:
1928:
474:
437:
6490:
6340:
6093:
6047:
5461:
5035:
4775:
4759:
3052:
2649:
2050:
1352:
1091:
1083:
1055:
1050:
1041:
998:
940:
737:
281:
155:
4213:
3231:
3109:
4755:
4698:
4458:
4416:
4380:
4141:
4074:
3073:
1958:
1895:
1509:
of all polynomials in one variable with integer coefficients is an integral domain; so is the ring
1310:
1263:
1109:
1099:
950:
850:
842:
833:
817:
563:
371:
322:
267:
161:
147:
75:
43:
4673:
3766:{\displaystyle \mathbb {C} \times \mathbb {C} \to \mathbb {C} \otimes _{\mathbb {R} }\mathbb {C} }
3639:
1997:
1873:
1512:
1481:
1362:
1316:
1283:
190:
6432:
4995:
4786:
4767:
4611:
4605:
4063:
3997:
3042:
2065:
1267:
915:
906:
864:
769:
683:
576:
134:
62:
2395:
6424:
6374:
6356:
6299:
6272:
6249:
6230:
6190:
6178:
6156:
6123:
6101:
6070:
6051:
6028:
6002:
5822:
4779:
4167:
4134:
Both notions of irreducible elements and prime elements generalize the ordinary definition of
3056:
2932:
2054:
1478:
are integral domains if the coefficients come from an integral domain. For instance, the ring
617:
414:
179:
120:
5655:
5622:
4166:
Every prime element is irreducible. The converse is not true in general: for example, in the
6448:
6440:
6416:
6402:
6390:
6382:
6364:
6348:
6182:
6133:
6016:
4751:
2840:
2162:
1847:
1184:
935:
811:
775:
675:
623:
609:
423:
365:
328:
128:
101:
87:
6204:
6170:
5715:
5688:
4889:
6444:
6386:
6200:
6166:
6152:
6137:
6024:
5567:
4790:
4771:
4693:
4558:
4453:(in any number of indeterminates) are integral domains. This is in particular the case if
4446:
4120:
3011:
2743:
2168:
1027:
1021:
1008:
988:
979:
882:
385:
173:
5596:
2611:
2585:
2096:
503:
6344:
4328:
without dividing either factor). In a unique factorization domain (or more generally, a
3391:
6288:
5441:
5381:
5142:{\displaystyle A\otimes _{k}B\to A/{\mathfrak {m}}\otimes _{k}B=k\otimes _{k}B\simeq B}
3371:
3351:
2912:
2723:
2703:
2683:
2663:
2257:
2237:
2217:
2197:
1992:
1967:
1827:
1672:
1571:
1069:
429:
17:
6369:
6326:
6479:
4793:
4424:
4087:
3077:
1985:
1920:
1340:
955:
920:
877:
763:
570:
466:
81:
1227:
An integral domain is a nonzero commutative ring in which for every nonzero element
1216:
An integral domain is a ring for which the set of nonzero elements is a commutative
1209:
An integral domain is a nonzero commutative ring for which every nonzero element is
4809:
4733:
4135:
2389:
1356:
1344:
1251:, so it is equivalent to require that every nonzero element of the ring be regular.
1192:
1129:
1060:
894:
602:
377:
273:
4721:
4435:
4128:
3059:) in general. The only case where this algebraic set may be irreducible is when
3038:
2793:
2638:
1988:
1336:
1203:
1119:
1114:
1003:
993:
967:
960:
664:
582:
293:
167:
49:
6332:
Proceedings of the
National Academy of Sciences of the United States of America
4005:
are the elements that divide 1; these are precisely the invertible elements in
6268:
6144:
6115:
4329:
1475:
1255:
1199:
869:
793:
754:
Some specific kinds of integral domains are given with the following chain of
744:
347:
6428:
6360:
3629:{\displaystyle e_{2}={\tfrac {1}{2}}(1\otimes 1)+{\tfrac {1}{2}}(i\otimes i)}
3542:{\displaystyle e_{1}={\tfrac {1}{2}}(1\otimes 1)-{\tfrac {1}{2}}(i\otimes i)}
4837:
4785:
More generally, a commutative ring is an integral domain if and only if its
4078:
is a nonzero non-unit that cannot be written as a product of two non-units.
2827:{\displaystyle M=N=({\begin{smallmatrix}0&1\\0&0\end{smallmatrix}})}
2653:
2090:
1240:
1187:
in which the product of any two nonzero elements is nonzero. Equivalently:
1181:
1124:
930:
887:
855:
672:
307:
212:
6378:
5747:-algebras that are integral domains and thus, using the previous property,
1343:
integral domain is a field. In particular, all finite integral domains are
6452:
6394:
6352:
6468:
5225:{\textstyle \sum {\overline {f_{i}}}g_{i}\sum {\overline {h_{i}}}g_{i}=0}
2058:
925:
301:
287:
33:
6436:
5557:{\textstyle (\sum f_{i}A)(\sum h_{i}A)\subset \operatorname {Jac} (A)=}
1303:
1259:
687:
185:
69:
4523:
The cancellation property holds for ideals in any integral domain: if
4321:{\displaystyle \left(2+{\sqrt {-5}}\right)\left(2-{\sqrt {-5}}\right)}
1217:
859:
6420:
4852:– the extra structure needed for an integral domain to be principal
6151:. Graduate Texts in Mathematics. Vol. 211. Berlin, New York:
4732:
Integral domains are characterized by the condition that they are
1191:
An integral domain is a nonzero commutative ring with no nonzero
4464:
The cancellation property holds in any integral domain: for any
3711:{\displaystyle \mathbb {C} \otimes _{\mathbb {R} }\mathbb {C} }
3451:{\displaystyle \mathbb {C} \otimes _{\mathbb {R} }\mathbb {C} }
4985:{\textstyle \sum f_{i}\otimes g_{i}\sum h_{j}\otimes g_{j}=0}
4778:
is an integral domain if and only if the algebraic set is an
2366:
2327:
2291:
1675:
is an integral domain. Integrality can be checked by showing
5812:{\displaystyle A\otimes _{k}B=\varinjlim A_{i}\otimes _{k}B}
4652:" in the sense that there is an injective ring homomorphism
2025:
3328:
3206:
4259:
has no integer solutions), but not prime (since 3 divides
3041:. The geometric interpretation of this result is that the
2068:
is an integral domain. In fact, a regular local ring is a
5593:
4572:
are integral domains over an algebraically closed field
1817:{\displaystyle \mathbb {Z} /(x^{2}-n)\cong \mathbb {Z} }
4550:
An integral domain is equal to the intersection of its
3893:{\displaystyle z\otimes w\mapsto (zw,z{\overline {w}})}
740:" for the general case including noncommutative rings.
5658:
5625:
5488:
5291:
5238:
5155:
4923:
4400:
is an integral domain if and only if the ideal (0) of
3909:
Divisibility, prime elements, and irreducible elements
3904:
of irreducible affine schemes need not be irreducible.
3836:{\displaystyle (z,w)\mapsto z\cdot e_{1}+w\cdot e_{2}}
3600:
3570:
3513:
3483:
3303:
3257:
3181:
3144:
1266:. (Given an integral domain, one can embed it in its
1198:
An integral domain is a commutative ring in which the
27:
Commutative ring with no zero divisors other than zero
5825:
5753:
5718:
5691:
5599:
5570:
5464:
5444:
5404:
5384:
5344:
5066:
5038:
4998:
4892:
4701:
4676:
4345:
4265:
4216:
4176:
4144:
3849:
3779:
3724:
3685:
3642:
3555:
3468:
3425:
3394:
3374:
3354:
3088:
3014:
2935:
2915:
2843:
2775:
2746:
2726:
2706:
2686:
2666:
2614:
2588:
2560:
2495:
2424:
2398:
2392:
of two nonzero commutative rings. In such a product
2352:
2316:
2280:
2260:
2240:
2220:
2200:
2171:
2128:
2099:
2053:
is an integral domain. The same is true for rings of
2022:
2000:
1970:
1931:
1898:
1876:
1850:
1830:
1751:
1681:
1583:
1515:
1484:
1389:
1365:
1319:
1286:
515:
477:
440:
220:
193:
6225:
Milies, César
Polcino; Sehgal, Sudarshan K. (2002).
2634:
are nonzero, while their product is 0 in this ring.
6469:"where does the term "integral domain" come from?"
6320:, vol. 1, Berlin, Heidelberg: Springer-Verlag
6287:
6122:(Third ed.), Reading, Mass.: Addison-Wesley,
6001:. University Mathematical Texts. Oliver and Boyd.
5929:
5831:
5811:
5731:
5704:
5677:
5644:
5611:
5585:
5556:
5474:
5450:
5430:
5390:
5370:
5330:
5277:
5224:
5141:
5048:
5024:
4984:
4905:
4712:
4684:
4371:
4320:
4251:
4202:
4155:
3892:
3835:
3765:
3718:is not a domain. In fact, there is an isomorphism
3710:
3671:
3628:
3541:
3450:
3403:
3380:
3360:
3334:
3029:
2988:
2921:
2901:
2826:
2761:
2732:
2712:
2692:
2672:
2626:
2600:
2574:
2546:
2478:
2410:
2377:
2338:
2302:
2266:
2246:
2226:
2206:
2186:
2149:
2111:
2041:
2008:
1976:
1946:
1909:
1884:
1862:
1836:
1816:
1730:
1663:
1558:
1501:
1462:
1373:
1327:
1294:
680:the product of any two nonzero elements is nonzero
539:
492:
455:
247:
201:
4670:. The field of fractions of the ring of integers
4506:. Another way to state this is that the function
1824:is an integral domain for any non-square integer
4591:is an integral domain. This is a consequence of
1664:{\displaystyle \mathbb {C} /(y^{2}-x(x-1)(x-2))}
694:. In an integral domain, every nonzero element
5882:
4662:such that any injective ring homomorphism from
4163:if one considers as prime the negative primes.
1220:under multiplication (because a monoid must be
4819:is an integral domain of prime characteristic
4332:), an irreducible element is a prime element.
1313:is an integral domain. For example, the field
682:. Integral domains are generalizations of the
6327:"Unique factorization in regular local rings"
1152:
644:
8:
6042:Dummit, David S.; Foote, Richard M. (2004).
5743:is an inductive limit of finitely generated
5331:{\textstyle \sum {\overline {h_{i}}}g_{i}=0}
5278:{\textstyle \sum {\overline {f_{i}}}g_{i}=0}
1961:of an integral domain is an integral domain.
248:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
5906:
5870:
690:and provide a natural setting for studying
4561:of integral domains is an integral domain.
2547:{\displaystyle \mathbb {Z} /(x^{2}-n^{2})}
2150:{\displaystyle \mathbb {Z} /m\mathbb {Z} }
1339:is an integral domain. Conversely, every
1159:
1145:
829:
651:
637:
38:
6368:
5824:
5800:
5790:
5773:
5761:
5752:
5723:
5717:
5696:
5690:
5666:
5657:
5633:
5624:
5598:
5569:
5521:
5499:
5487:
5466:
5465:
5463:
5443:
5411:
5405:
5403:
5383:
5351:
5345:
5343:
5316:
5301:
5295:
5290:
5263:
5248:
5242:
5237:
5210:
5195:
5189:
5180:
5165:
5159:
5154:
5124:
5105:
5095:
5094:
5089:
5074:
5065:
5040:
5039:
5037:
5016:
5003:
4997:
4970:
4957:
4944:
4931:
4922:
4897:
4891:
4758:). The former condition ensures that the
4703:
4702:
4700:
4678:
4677:
4675:
4355:
4347:
4346:
4344:
4303:
4277:
4264:
4237:
4221:
4215:
4186:
4178:
4177:
4175:
4146:
4145:
4143:
3877:
3848:
3827:
3808:
3778:
3759:
3758:
3752:
3751:
3750:
3742:
3741:
3734:
3733:
3726:
3725:
3723:
3704:
3703:
3697:
3696:
3695:
3687:
3686:
3684:
3657:
3647:
3641:
3599:
3569:
3560:
3554:
3512:
3482:
3473:
3467:
3444:
3443:
3437:
3436:
3435:
3427:
3426:
3424:
3393:
3373:
3353:
3302:
3256:
3226:
3180:
3143:
3104:
3087:
3055:that is not irreducible (that is, not an
3013:
2977:
2958:
2934:
2914:
2882:
2873:
2854:
2842:
2791:
2774:
2745:
2725:
2705:
2685:
2665:
2613:
2587:
2568:
2567:
2559:
2535:
2522:
2510:
2497:
2496:
2494:
2423:
2397:
2369:
2365:
2351:
2339:{\displaystyle y\not \equiv 0{\bmod {m}}}
2330:
2326:
2315:
2303:{\displaystyle x\not \equiv 0{\bmod {m}}}
2294:
2290:
2279:
2259:
2239:
2219:
2199:
2170:
2143:
2142:
2134:
2130:
2129:
2127:
2098:
2024:
2023:
2021:
2002:
2001:
1999:
1969:
1938:
1934:
1933:
1930:
1900:
1899:
1897:
1878:
1877:
1875:
1849:
1829:
1804:
1797:
1796:
1778:
1766:
1753:
1752:
1750:
1686:
1680:
1616:
1604:
1585:
1584:
1582:
1547:
1528:
1517:
1516:
1514:
1486:
1485:
1483:
1450:
1449:
1437:
1426:
1425:
1419:
1402:
1401:
1391:
1390:
1388:
1367:
1366:
1364:
1321:
1320:
1318:
1288:
1287:
1285:
540:{\displaystyle \mathbb {Z} (p^{\infty })}
528:
517:
516:
514:
484:
480:
479:
476:
447:
443:
442:
439:
241:
240:
232:
228:
227:
219:
195:
194:
192:
5894:
5858:
2165:. Indeed, choose a proper factorization
1870:, then this ring is always a subring of
5851:
5564:the intersection of all maximal ideals
4867:
4808:of an integral domain is either 0 or a
3069:, which defines the same algebraic set.
832:
41:
6088:, London: Blaisdell Publishing Company
5951:
5940:
5619:is a prime ideal, this implies either
2479:{\displaystyle (1,0)\cdot (0,1)=(0,0)}
2057:on connected open subsets of analytic
1231:, the function that maps each element
800:unique factorization domains
6325:Auslander, M; Buchsbaum, D A (1959).
6069:(3rd ed.). John Wiley and Sons.
5431:{\displaystyle {\overline {h_{i}}}=0}
5371:{\displaystyle {\overline {f_{i}}}=0}
5032:are nonzero). For each maximal ideal
4720:The field of fractions of a field is
4430:is an integral domain if and only if
2378:{\displaystyle xy\equiv 0{\bmod {m}}}
1280:The archetypical example is the ring
1254:An integral domain is a ring that is
7:
5917:
3636:. They are orthogonal, meaning that
2700:are matrices such that the image of
108:Free product of associative algebras
5467:
5096:
5041:
4379:, there is unique factorization of
4009:. Units divide all other elements.
788:integrally closed domains
6265:Algebra: groups, rings, and fields
6100:(3rd ed.). Cengage Learning.
529:
25:
6415:(2) (published April 1958): 382.
6339:(5) (published May 1959): 733–4.
6098:Abstract Algebra: An Introduction
5060:, consider the ring homomorphism
4372:{\displaystyle \mathbb {Z} \left}
4203:{\displaystyle \mathbb {Z} \left}
2929:and any non-constant polynomials
2792:
2769:. For example, this happens for
2575:{\displaystyle n\in \mathbb {Z} }
2042:{\displaystyle {\mathcal {H}}(U)}
1731:{\displaystyle y^{2}-x(x-1)(x-2)}
806:principal ideal domains
596:Noncommutative algebraic geometry
6314:van der Waerden, Bartel Leendert
5685:is the zero ideal; i.e., either
4800:Characteristic and homomorphisms
4445:be an integral domain. Then the
3458:. This ring has two non-trivial
1947:{\displaystyle \mathbb {Z} _{p}}
1892:, otherwise, it is a subring of
824:algebraically closed fields
493:{\displaystyle \mathbb {Q} _{p}}
456:{\displaystyle \mathbb {Z} _{p}}
6408:American Journal of Mathematics
6218:Graduate Studies in Mathematics
6214:Noncommutative Noetherian Rings
6212:McConnell, J.C.; Robson, J.C.,
6067:Modern Algebra: An Introduction
5475:{\displaystyle {\mathfrak {m}}}
5049:{\displaystyle {\mathfrak {m}}}
3210:
6227:An introduction to group rings
6189:. New York: The Macmillan Co.
5930:Auslander & Buchsbaum 1959
5606:
5600:
5580:
5574:
5548:
5542:
5530:
5511:
5508:
5489:
5083:
4252:{\displaystyle a^{2}+5b^{2}=3}
3887:
3862:
3859:
3795:
3792:
3780:
3738:
3623:
3611:
3593:
3581:
3536:
3524:
3506:
3494:
3220:
3214:
3098:
3092:
3024:
3015:
2983:
2951:
2896:
2887:
2879:
2847:
2821:
2788:
2720:is contained in the kernel of
2541:
2515:
2507:
2501:
2473:
2461:
2455:
2443:
2437:
2425:
2036:
2030:
1811:
1801:
1790:
1771:
1763:
1757:
1725:
1713:
1710:
1698:
1658:
1655:
1643:
1640:
1628:
1609:
1601:
1589:
1553:
1521:
1496:
1490:
1247:with this property are called
534:
521:
1:
5338:and, by linear independence,
4713:{\displaystyle \mathbb {Q} .}
4516:is injective for any nonzero
4156:{\displaystyle \mathbb {Z} ,}
3974:, if there exists an element
1910:{\displaystyle \mathbb {C} .}
6246:Concepts in abstract algebra
5999:Elementary rings and modules
5417:
5357:
5307:
5254:
5201:
5171:
4685:{\displaystyle \mathbb {Z} }
4119:is prime if and only if the
3900:. This example shows that a
3882:
3843:. Its inverse is defined by
3672:{\displaystyle e_{1}e_{2}=0}
2009:{\displaystyle \mathbb {C} }
1885:{\displaystyle \mathbb {R} }
1559:{\displaystyle \mathbb {C} }
1502:{\displaystyle \mathbb {Z} }
1374:{\displaystyle \mathbb {Z} }
1328:{\displaystyle \mathbb {R} }
1295:{\displaystyle \mathbb {Z} }
202:{\displaystyle \mathbb {Z} }
6263:Rowen, Louis Halle (1994).
5025:{\displaystyle f_{i},h_{j}}
4878:is finitely generated as a
4754:(that is there is only one
4666:to a field factors through
4115:. Equivalently, an element
1349:Wedderburn's little theorem
1235:of the ring to the product
354:Unique factorization domain
6507:
6296:Cambridge University Press
6046:(3rd ed.). New York:
4603:
4476:in an integral domain, if
4411:is a commutative ring and
3915:Divisibility (ring theory)
3912:
3080:. Consider the functions
114:Tensor product of algebras
31:
5997:Adamson, Iain T. (1972).
4770:, into the fact that the
4593:Hilbert's nullstellensatz
2411:{\displaystyle R\times S}
1671:corresponding to a plane
6244:Lanski, Charles (2005).
6065:Durbin, John R. (1993).
5832:{\displaystyle \square }
5678:{\textstyle \sum h_{i}A}
5645:{\textstyle \sum f_{i}A}
4621:is the set of fractions
3388:is everywhere zero, but
2989:{\displaystyle f,g\in k}
2081:The following rings are
1359:). The ring of integers
392:Formal power series ring
342:Integrally closed domain
32:Not to be confused with
6290:Rings and factorization
6084:Herstein, I.N. (1964),
5871:Dummit & Foote 2004
5819:is an integral domain.
5739:are all zero. Finally,
3923:is an integral domain.
734:multiplicative identity
401:Algebraic number theory
94:Total ring of fractions
18:Associate (ring theory)
6286:Sharpe, David (1987).
5907:McConnell & Robson
5833:
5813:
5733:
5706:
5679:
5646:
5613:
5587:
5558:
5482:is arbitrary, we have
5476:
5452:
5432:
5392:
5372:
5332:
5279:
5226:
5143:
5050:
5026:
4986:
4907:
4825:Frobenius endomorphism
4714:
4686:
4617:of an integral domain
4385:Lasker–Noether theorem
4373:
4322:
4253:
4204:
4157:
3894:
3837:
3767:
3712:
3673:
3630:
3543:
3452:
3405:
3382:
3362:
3336:
3067:irreducible polynomial
3031:
2990:
2923:
2903:
2902:{\displaystyle k/(fg)}
2828:
2763:
2734:
2714:
2694:
2674:
2628:
2602:
2576:
2548:
2480:
2412:
2379:
2340:
2304:
2268:
2248:
2228:
2208:
2188:
2151:
2113:
2043:
2010:
1978:
1954:is an integral domain.
1948:
1911:
1886:
1864:
1863:{\displaystyle n>0}
1838:
1818:
1740:irreducible polynomial
1732:
1665:
1566:of all polynomials in
1560:
1503:
1464:
1375:
1329:
1296:
1224:under multiplication).
812:Euclidean domains
776:commutative rings
743:Some sources, notably
558:Noncommutative algebra
541:
494:
457:
409:Algebraic number field
360:Principal ideal domain
249:
203:
141:Frobenius endomorphism
6353:10.1073/PNAS.45.5.733
6094:Hungerford, Thomas W.
6021:Algebra, Chapters 1–3
5834:
5814:
5734:
5732:{\displaystyle h_{i}}
5707:
5705:{\displaystyle f_{i}}
5680:
5647:
5614:
5588:
5559:
5477:
5453:
5433:
5393:
5373:
5333:
5280:
5227:
5144:
5051:
5027:
4987:
4908:
4906:{\displaystyle g_{i}}
4856:Zero-product property
4724:to the field itself.
4715:
4687:
4374:
4323:
4254:
4205:
4158:
3895:
3838:
3768:
3713:
3674:
3631:
3544:
3453:
3406:
3383:
3363:
3337:
3032:
2991:
2924:
2904:
2829:
2764:
2735:
2715:
2695:
2675:
2629:
2603:
2577:
2549:
2481:
2413:
2380:
2341:
2305:
2269:
2249:
2229:
2209:
2189:
2152:
2114:
2051:holomorphic functions
2044:
2011:
1979:
1949:
1912:
1887:
1865:
1839:
1819:
1733:
1666:
1561:
1504:
1465:
1376:
1330:
1297:
1213:under multiplication.
782:integral domains
751:for integral domain.
700:cancellation property
542:
495:
458:
250:
204:
34:domain of integration
6023:. Berlin, New York:
5954:, p. 224, "Elements
5883:van der Waerden 1966
5823:
5751:
5716:
5689:
5656:
5623:
5597:
5586:{\displaystyle =(0)}
5568:
5486:
5462:
5442:
5402:
5382:
5342:
5289:
5236:
5153:
5149:. Then the image is
5064:
5036:
4996:
4992:(only finitely many
4921:
4890:
4882:-algebra and pick a
4874:Proof: First assume
4776:affine algebraic set
4766:This translates, in
4699:
4674:
4343:
4337:unique factorization
4263:
4214:
4174:
4142:
3847:
3777:
3722:
3683:
3640:
3553:
3466:
3423:
3392:
3372:
3352:
3086:
3074:continuous functions
3053:affine algebraic set
3030:{\displaystyle (fg)}
3012:
2933:
2913:
2841:
2773:
2762:{\displaystyle MN=0}
2744:
2724:
2704:
2684:
2664:
2612:
2586:
2558:
2493:
2422:
2396:
2350:
2314:
2278:
2258:
2238:
2218:
2198:
2187:{\displaystyle m=xy}
2169:
2126:
2097:
2020:
1998:
1968:
1929:
1896:
1874:
1848:
1828:
1749:
1679:
1581:
1513:
1482:
1387:
1363:
1347:(more generally, by
1317:
1284:
1056:Group with operators
999:Complemented lattice
834:Algebraic structures
564:Noncommutative rings
513:
475:
438:
282:Non-associative ring
218:
191:
148:Algebraic structures
6486:Commutative algebra
6345:1959PNAS...45..733A
6220:, vol. 30, AMS
5612:{\displaystyle (0)}
4850:Dedekind–Hasse norm
4756:minimal prime ideal
4396:A commutative ring
4081:A nonzero non-unit
4075:irreducible element
4038:associated elements
2627:{\displaystyle x-n}
2601:{\displaystyle x+n}
2112:{\displaystyle 0=1}
2093:(the ring in which
1959:formal power series
1110:Composition algebra
870:Quasigroup and loop
323:Commutative algebra
162:Associative algebra
44:Algebraic structure
6179:Mac Lane, Saunders
5829:
5809:
5781:
5729:
5702:
5675:
5642:
5609:
5583:
5554:
5472:
5448:
5428:
5388:
5368:
5328:
5275:
5222:
5139:
5046:
5022:
4982:
4903:
4768:algebraic geometry
4728:Algebraic geometry
4710:
4682:
4612:field of fractions
4606:Field of fractions
4600:Field of fractions
4554:at maximal ideals.
4369:
4318:
4249:
4200:
4153:
4095:divides a product
4052:are associates if
3890:
3833:
3763:
3708:
3669:
3626:
3609:
3579:
3539:
3522:
3492:
3448:
3404:{\displaystyle fg}
3401:
3378:
3358:
3332:
3327:
3312:
3266:
3205:
3190:
3153:
3027:
2986:
2919:
2899:
2837:The quotient ring
2824:
2819:
2818:
2759:
2730:
2710:
2690:
2670:
2624:
2598:
2572:
2544:
2489:The quotient ring
2476:
2408:
2375:
2336:
2300:
2264:
2244:
2224:
2204:
2184:
2147:
2122:The quotient ring
2109:
2085:integral domains.
2066:regular local ring
2055:analytic functions
2049:consisting of all
2039:
2006:
1974:
1944:
1907:
1882:
1860:
1834:
1814:
1728:
1661:
1556:
1499:
1460:
1371:
1325:
1292:
1268:field of fractions
577:Semiprimitive ring
537:
490:
453:
261:Related structures
245:
199:
135:Inner automorphism
121:Ring homomorphisms
6403:Nagata, Masayoshi
6248:. AMS Bookstore.
6183:Birkhoff, Garrett
6162:978-0-387-95385-4
6129:978-0-201-55540-0
6107:978-1-111-56962-4
6086:Topics in Algebra
6057:978-0-471-43334-7
6034:978-3-540-64243-5
6017:Bourbaki, Nicolas
5774:
5451:{\displaystyle i}
5420:
5391:{\displaystyle i}
5360:
5310:
5257:
5204:
5174:
4780:algebraic variety
4404:is a prime ideal.
4363:
4339:does not hold in
4311:
4285:
4194:
4168:quadratic integer
4044:. Equivalently,
3919:In this section,
3885:
3608:
3578:
3521:
3491:
3381:{\displaystyle g}
3361:{\displaystyle f}
3311:
3265:
3189:
3152:
3065:is a power of an
3057:algebraic variety
2922:{\displaystyle k}
2733:{\displaystyle M}
2713:{\displaystyle N}
2693:{\displaystyle N}
2673:{\displaystyle M}
2582:. The images of
2267:{\displaystyle m}
2247:{\displaystyle 1}
2234:are not equal to
2227:{\displaystyle y}
2207:{\displaystyle x}
1977:{\displaystyle U}
1837:{\displaystyle n}
1809:
1169:
1168:
661:
660:
618:Geometric algebra
329:Commutative rings
180:Category of rings
16:(Redirected from
6498:
6472:
6456:
6398:
6372:
6321:
6309:
6293:
6282:
6259:
6240:
6221:
6208:
6174:
6140:
6111:
6089:
6080:
6061:
6044:Abstract Algebra
6038:
6012:
5983:
5949:
5943:
5938:
5932:
5927:
5921:
5920:, pp. 91–92
5915:
5909:
5904:
5898:
5897:, pp. 88–90
5892:
5886:
5880:
5874:
5868:
5862:
5856:
5839:
5838:
5836:
5835:
5830:
5818:
5816:
5815:
5810:
5805:
5804:
5795:
5794:
5782:
5766:
5765:
5738:
5736:
5735:
5730:
5728:
5727:
5712:are all zero or
5711:
5709:
5708:
5703:
5701:
5700:
5684:
5682:
5681:
5676:
5671:
5670:
5651:
5649:
5648:
5643:
5638:
5637:
5618:
5616:
5615:
5610:
5592:
5590:
5589:
5584:
5563:
5561:
5560:
5555:
5526:
5525:
5504:
5503:
5481:
5479:
5478:
5473:
5471:
5470:
5457:
5455:
5454:
5449:
5437:
5435:
5434:
5429:
5421:
5416:
5415:
5406:
5397:
5395:
5394:
5389:
5377:
5375:
5374:
5369:
5361:
5356:
5355:
5346:
5337:
5335:
5334:
5329:
5321:
5320:
5311:
5306:
5305:
5296:
5284:
5282:
5281:
5276:
5268:
5267:
5258:
5253:
5252:
5243:
5232:and thus either
5231:
5229:
5228:
5223:
5215:
5214:
5205:
5200:
5199:
5190:
5185:
5184:
5175:
5170:
5169:
5160:
5148:
5146:
5145:
5140:
5129:
5128:
5110:
5109:
5100:
5099:
5093:
5079:
5078:
5055:
5053:
5052:
5047:
5045:
5044:
5031:
5029:
5028:
5023:
5021:
5020:
5008:
5007:
4991:
4989:
4988:
4983:
4975:
4974:
4962:
4961:
4949:
4948:
4936:
4935:
4912:
4910:
4909:
4904:
4902:
4901:
4872:
4835:
4749:
4742:
4719:
4717:
4716:
4711:
4706:
4694:rational numbers
4692:is the field of
4691:
4689:
4688:
4683:
4681:
4661:
4647:
4590:
4546:
4532:
4515:
4505:
4495:
4485:
4447:polynomial rings
4378:
4376:
4375:
4370:
4368:
4364:
4356:
4350:
4327:
4325:
4324:
4319:
4317:
4313:
4312:
4304:
4291:
4287:
4286:
4278:
4258:
4256:
4255:
4250:
4242:
4241:
4226:
4225:
4209:
4207:
4206:
4201:
4199:
4195:
4187:
4181:
4162:
4160:
4159:
4154:
4149:
4061:
3991:
3938:, one says that
3899:
3897:
3896:
3891:
3886:
3878:
3842:
3840:
3839:
3834:
3832:
3831:
3813:
3812:
3772:
3770:
3769:
3764:
3762:
3757:
3756:
3755:
3745:
3737:
3729:
3717:
3715:
3714:
3709:
3707:
3702:
3701:
3700:
3690:
3678:
3676:
3675:
3670:
3662:
3661:
3652:
3651:
3635:
3633:
3632:
3627:
3610:
3601:
3580:
3571:
3565:
3564:
3548:
3546:
3545:
3540:
3523:
3514:
3493:
3484:
3478:
3477:
3457:
3455:
3454:
3449:
3447:
3442:
3441:
3440:
3430:
3410:
3408:
3407:
3402:
3387:
3385:
3384:
3379:
3367:
3365:
3364:
3359:
3341:
3339:
3338:
3333:
3331:
3330:
3324:
3320:
3313:
3304:
3272:
3268:
3267:
3258:
3209:
3208:
3202:
3198:
3191:
3182:
3159:
3155:
3154:
3145:
3064:
3050:
3036:
3034:
3033:
3028:
3007:
3001:
2996:. The images of
2995:
2993:
2992:
2987:
2982:
2981:
2963:
2962:
2928:
2926:
2925:
2920:
2908:
2906:
2905:
2900:
2886:
2878:
2877:
2859:
2858:
2833:
2831:
2830:
2825:
2820:
2768:
2766:
2765:
2760:
2739:
2737:
2736:
2731:
2719:
2717:
2716:
2711:
2699:
2697:
2696:
2691:
2679:
2677:
2676:
2671:
2633:
2631:
2630:
2625:
2607:
2605:
2604:
2599:
2581:
2579:
2578:
2573:
2571:
2553:
2551:
2550:
2545:
2540:
2539:
2527:
2526:
2514:
2500:
2485:
2483:
2482:
2477:
2417:
2415:
2414:
2409:
2384:
2382:
2381:
2376:
2374:
2373:
2345:
2343:
2342:
2337:
2335:
2334:
2309:
2307:
2306:
2301:
2299:
2298:
2273:
2271:
2270:
2265:
2253:
2251:
2250:
2245:
2233:
2231:
2230:
2225:
2213:
2211:
2210:
2205:
2193:
2191:
2190:
2185:
2163:composite number
2156:
2154:
2153:
2148:
2146:
2138:
2133:
2118:
2116:
2115:
2110:
2048:
2046:
2045:
2040:
2029:
2028:
2016:, then the ring
2015:
2013:
2012:
2007:
2005:
1983:
1981:
1980:
1975:
1953:
1951:
1950:
1945:
1943:
1942:
1937:
1916:
1914:
1913:
1908:
1903:
1891:
1889:
1888:
1883:
1881:
1869:
1867:
1866:
1861:
1843:
1841:
1840:
1835:
1823:
1821:
1820:
1815:
1810:
1805:
1800:
1783:
1782:
1770:
1756:
1737:
1735:
1734:
1729:
1691:
1690:
1670:
1668:
1667:
1662:
1621:
1620:
1608:
1588:
1570:-variables with
1565:
1563:
1562:
1557:
1552:
1551:
1533:
1532:
1520:
1508:
1506:
1505:
1500:
1489:
1469:
1467:
1466:
1461:
1453:
1448:
1447:
1429:
1424:
1423:
1405:
1394:
1380:
1378:
1377:
1372:
1370:
1334:
1332:
1331:
1326:
1324:
1301:
1299:
1298:
1293:
1291:
1185:commutative ring
1161:
1154:
1147:
936:Commutative ring
865:Rack and quandle
830:
794:GCD domains
756:class inclusions
728:
718:
708:
676:commutative ring
653:
646:
639:
624:Operator algebra
610:Clifford algebra
546:
544:
543:
538:
533:
532:
520:
499:
497:
496:
491:
489:
488:
483:
462:
460:
459:
454:
452:
451:
446:
424:Ring of integers
418:
415:Integers modulo
366:Euclidean domain
254:
252:
251:
246:
244:
236:
231:
208:
206:
205:
200:
198:
102:Product of rings
88:Fractional ideal
47:
39:
21:
6506:
6505:
6501:
6500:
6499:
6497:
6496:
6495:
6476:
6475:
6467:
6464:
6459:
6421:10.2307/2372791
6401:
6324:
6312:
6306:
6285:
6279:
6262:
6256:
6243:
6237:
6224:
6211:
6197:
6177:
6163:
6153:Springer-Verlag
6143:
6130:
6114:
6108:
6092:
6083:
6077:
6064:
6058:
6041:
6035:
6025:Springer-Verlag
6015:
6009:
5996:
5992:
5987:
5986:
5962:of are called
5950:
5946:
5939:
5935:
5928:
5924:
5916:
5912:
5905:
5901:
5893:
5889:
5881:
5877:
5869:
5865:
5857:
5853:
5848:
5843:
5842:
5821:
5820:
5796:
5786:
5757:
5749:
5748:
5719:
5714:
5713:
5692:
5687:
5686:
5662:
5654:
5653:
5629:
5621:
5620:
5595:
5594:
5566:
5565:
5517:
5495:
5484:
5483:
5460:
5459:
5440:
5439:
5407:
5400:
5399:
5380:
5379:
5347:
5340:
5339:
5312:
5297:
5287:
5286:
5259:
5244:
5234:
5233:
5206:
5191:
5176:
5161:
5151:
5150:
5120:
5101:
5070:
5062:
5061:
5034:
5033:
5012:
4999:
4994:
4993:
4966:
4953:
4940:
4927:
4919:
4918:
4893:
4888:
4887:
4873:
4869:
4864:
4846:
4827:
4802:
4772:coordinate ring
4744:
4737:
4730:
4697:
4696:
4672:
4671:
4653:
4642:
4608:
4602:
4586:
4577:
4559:inductive limit
4538:
4524:
4507:
4497:
4487:
4477:
4393:
4351:
4341:
4340:
4296:
4292:
4270:
4266:
4261:
4260:
4233:
4217:
4212:
4211:
4182:
4172:
4171:
4140:
4139:
4127:) is a nonzero
4121:principal ideal
4053:
3983:
3926:Given elements
3917:
3911:
3845:
3844:
3823:
3804:
3775:
3774:
3746:
3720:
3719:
3691:
3681:
3680:
3653:
3643:
3638:
3637:
3556:
3551:
3550:
3469:
3464:
3463:
3431:
3421:
3420:
3390:
3389:
3370:
3369:
3350:
3349:
3326:
3325:
3301:
3297:
3289:
3274:
3273:
3249:
3245:
3237:
3227:
3204:
3203:
3179:
3175:
3167:
3161:
3160:
3136:
3132:
3124:
3105:
3084:
3083:
3060:
3046:
3010:
3009:
3003:
2997:
2973:
2954:
2931:
2930:
2911:
2910:
2869:
2850:
2839:
2838:
2817:
2816:
2811:
2805:
2804:
2799:
2771:
2770:
2742:
2741:
2722:
2721:
2702:
2701:
2682:
2681:
2662:
2661:
2610:
2609:
2584:
2583:
2556:
2555:
2531:
2518:
2491:
2490:
2420:
2419:
2394:
2393:
2348:
2347:
2312:
2311:
2276:
2275:
2256:
2255:
2236:
2235:
2216:
2215:
2196:
2195:
2167:
2166:
2124:
2123:
2095:
2094:
2079:
2018:
2017:
1996:
1995:
1966:
1965:
1932:
1927:
1926:
1894:
1893:
1872:
1871:
1846:
1845:
1826:
1825:
1774:
1747:
1746:
1682:
1677:
1676:
1612:
1579:
1578:
1543:
1524:
1511:
1510:
1480:
1479:
1433:
1415:
1385:
1384:
1361:
1360:
1315:
1314:
1282:
1281:
1277:
1178:integral domain
1174:
1165:
1136:
1135:
1134:
1105:Non-associative
1087:
1076:
1075:
1065:
1045:
1034:
1033:
1022:Map of lattices
1018:
1014:Boolean algebra
1009:Heyting algebra
983:
972:
971:
965:
946:Integral domain
910:
899:
898:
892:
846:
747:, use the term
720:
710:
703:
669:integral domain
657:
628:
627:
560:
550:
549:
524:
511:
510:
478:
473:
472:
441:
436:
435:
416:
386:Polynomial ring
336:Integral domain
325:
315:
314:
216:
215:
189:
188:
174:Involutive ring
59:
48:
42:
37:
28:
23:
22:
15:
12:
11:
5:
6504:
6502:
6494:
6493:
6488:
6478:
6477:
6474:
6473:
6463:
6462:External links
6460:
6458:
6457:
6399:
6322:
6310:
6304:
6283:
6277:
6260:
6254:
6241:
6235:
6222:
6209:
6195:
6175:
6161:
6141:
6128:
6112:
6106:
6090:
6081:
6075:
6062:
6056:
6039:
6033:
6013:
6007:
5993:
5991:
5988:
5985:
5984:
5944:
5933:
5922:
5910:
5899:
5887:
5875:
5863:
5850:
5849:
5847:
5844:
5841:
5840:
5828:
5808:
5803:
5799:
5793:
5789:
5785:
5780:
5777:
5772:
5769:
5764:
5760:
5756:
5726:
5722:
5699:
5695:
5674:
5669:
5665:
5661:
5641:
5636:
5632:
5628:
5608:
5605:
5602:
5582:
5579:
5576:
5573:
5553:
5550:
5547:
5544:
5541:
5538:
5535:
5532:
5529:
5524:
5520:
5516:
5513:
5510:
5507:
5502:
5498:
5494:
5491:
5469:
5447:
5427:
5424:
5419:
5414:
5410:
5387:
5367:
5364:
5359:
5354:
5350:
5327:
5324:
5319:
5315:
5309:
5304:
5300:
5294:
5274:
5271:
5266:
5262:
5256:
5251:
5247:
5241:
5221:
5218:
5213:
5209:
5203:
5198:
5194:
5188:
5183:
5179:
5173:
5168:
5164:
5158:
5138:
5135:
5132:
5127:
5123:
5119:
5116:
5113:
5108:
5104:
5098:
5092:
5088:
5085:
5082:
5077:
5073:
5069:
5043:
5019:
5015:
5011:
5006:
5002:
4981:
4978:
4973:
4969:
4965:
4960:
4956:
4952:
4947:
4943:
4939:
4934:
4930:
4926:
4900:
4896:
4866:
4865:
4863:
4860:
4859:
4858:
4853:
4845:
4842:
4806:characteristic
4801:
4798:
4729:
4726:
4709:
4705:
4680:
4604:Main article:
4601:
4598:
4597:
4596:
4582:
4562:
4555:
4548:
4533:, then either
4521:
4520:in the domain.
4462:
4439:
4405:
4392:
4389:
4367:
4362:
4359:
4354:
4349:
4316:
4310:
4307:
4302:
4299:
4295:
4290:
4284:
4281:
4276:
4273:
4269:
4248:
4245:
4240:
4236:
4232:
4229:
4224:
4220:
4198:
4193:
4190:
4185:
4180:
4152:
4148:
3910:
3907:
3906:
3905:
3889:
3884:
3881:
3876:
3873:
3870:
3867:
3864:
3861:
3858:
3855:
3852:
3830:
3826:
3822:
3819:
3816:
3811:
3807:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3761:
3754:
3749:
3744:
3740:
3736:
3732:
3728:
3706:
3699:
3694:
3689:
3668:
3665:
3660:
3656:
3650:
3646:
3625:
3622:
3619:
3616:
3613:
3607:
3604:
3598:
3595:
3592:
3589:
3586:
3583:
3577:
3574:
3568:
3563:
3559:
3538:
3535:
3532:
3529:
3526:
3520:
3517:
3511:
3508:
3505:
3502:
3499:
3496:
3490:
3487:
3481:
3476:
3472:
3446:
3439:
3434:
3429:
3418:tensor product
3413:
3412:
3400:
3397:
3377:
3357:
3345:
3344:
3343:
3342:
3329:
3323:
3319:
3316:
3310:
3307:
3300:
3296:
3293:
3290:
3288:
3285:
3282:
3279:
3276:
3275:
3271:
3264:
3261:
3255:
3252:
3248:
3244:
3241:
3238:
3236:
3233:
3232:
3230:
3225:
3222:
3219:
3216:
3213:
3207:
3201:
3197:
3194:
3188:
3185:
3178:
3174:
3171:
3168:
3166:
3163:
3162:
3158:
3151:
3148:
3142:
3139:
3135:
3131:
3128:
3125:
3123:
3120:
3117:
3114:
3111:
3110:
3108:
3103:
3100:
3097:
3094:
3091:
3070:
3026:
3023:
3020:
3017:
2985:
2980:
2976:
2972:
2969:
2966:
2961:
2957:
2953:
2950:
2947:
2944:
2941:
2938:
2918:
2909:for any field
2898:
2895:
2892:
2889:
2885:
2881:
2876:
2872:
2868:
2865:
2862:
2857:
2853:
2849:
2846:
2835:
2823:
2815:
2812:
2810:
2807:
2806:
2803:
2800:
2798:
2795:
2794:
2790:
2787:
2784:
2781:
2778:
2758:
2755:
2752:
2749:
2729:
2709:
2689:
2669:
2635:
2623:
2620:
2617:
2597:
2594:
2591:
2570:
2566:
2563:
2543:
2538:
2534:
2530:
2525:
2521:
2517:
2513:
2509:
2506:
2503:
2499:
2487:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2407:
2404:
2401:
2386:
2372:
2368:
2364:
2361:
2358:
2355:
2333:
2329:
2325:
2322:
2319:
2297:
2293:
2289:
2286:
2283:
2263:
2243:
2223:
2203:
2194:(meaning that
2183:
2180:
2177:
2174:
2145:
2141:
2137:
2132:
2120:
2108:
2105:
2102:
2078:
2075:
2074:
2073:
2062:
2038:
2035:
2032:
2027:
2004:
1973:
1962:
1955:
1941:
1936:
1924:-adic integers
1917:
1906:
1902:
1880:
1859:
1856:
1853:
1833:
1813:
1808:
1803:
1799:
1795:
1792:
1789:
1786:
1781:
1777:
1773:
1769:
1765:
1762:
1759:
1755:
1743:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1689:
1685:
1673:elliptic curve
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1619:
1615:
1611:
1607:
1603:
1600:
1597:
1594:
1591:
1587:
1575:
1555:
1550:
1546:
1542:
1539:
1536:
1531:
1527:
1523:
1519:
1498:
1495:
1492:
1488:
1472:
1471:
1470:
1459:
1456:
1452:
1446:
1443:
1440:
1436:
1432:
1428:
1422:
1418:
1414:
1411:
1408:
1404:
1400:
1397:
1393:
1369:
1323:
1307:
1290:
1276:
1273:
1272:
1271:
1252:
1225:
1214:
1207:
1196:
1173:
1170:
1167:
1166:
1164:
1163:
1156:
1149:
1141:
1138:
1137:
1133:
1132:
1127:
1122:
1117:
1112:
1107:
1102:
1096:
1095:
1094:
1088:
1082:
1081:
1078:
1077:
1074:
1073:
1070:Linear algebra
1064:
1063:
1058:
1053:
1047:
1046:
1040:
1039:
1036:
1035:
1032:
1031:
1028:Lattice theory
1024:
1017:
1016:
1011:
1006:
1001:
996:
991:
985:
984:
978:
977:
974:
973:
964:
963:
958:
953:
948:
943:
938:
933:
928:
923:
918:
912:
911:
905:
904:
901:
900:
891:
890:
885:
880:
874:
873:
872:
867:
862:
853:
847:
841:
840:
837:
836:
828:
827:
709:, an equality
702:, that is, if
659:
658:
656:
655:
648:
641:
633:
630:
629:
621:
620:
592:
591:
585:
579:
573:
561:
556:
555:
552:
551:
548:
547:
536:
531:
527:
523:
519:
500:
487:
482:
463:
450:
445:
433:-adic integers
426:
420:
411:
397:
396:
395:
394:
388:
382:
381:
380:
368:
362:
356:
350:
344:
326:
321:
320:
317:
316:
313:
312:
311:
310:
298:
297:
296:
290:
278:
277:
276:
258:
257:
256:
255:
243:
239:
235:
230:
226:
223:
209:
197:
176:
170:
164:
158:
144:
143:
137:
131:
117:
116:
110:
104:
98:
97:
96:
90:
78:
72:
60:
58:Basic concepts
57:
56:
53:
52:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6503:
6492:
6489:
6487:
6484:
6483:
6481:
6470:
6466:
6465:
6461:
6454:
6450:
6446:
6442:
6438:
6434:
6430:
6426:
6422:
6418:
6414:
6410:
6409:
6404:
6400:
6396:
6392:
6388:
6384:
6380:
6376:
6371:
6366:
6362:
6358:
6354:
6350:
6346:
6342:
6338:
6334:
6333:
6328:
6323:
6319:
6315:
6311:
6307:
6305:0-521-33718-6
6301:
6297:
6292:
6291:
6284:
6280:
6278:1-56881-028-8
6274:
6270:
6266:
6261:
6257:
6255:0-534-42323-X
6251:
6247:
6242:
6238:
6236:1-4020-0238-6
6232:
6228:
6223:
6219:
6215:
6210:
6206:
6202:
6198:
6196:1-56881-068-7
6192:
6188:
6184:
6180:
6176:
6172:
6168:
6164:
6158:
6154:
6150:
6146:
6142:
6139:
6135:
6131:
6125:
6121:
6117:
6113:
6109:
6103:
6099:
6095:
6091:
6087:
6082:
6078:
6076:0-471-51001-7
6072:
6068:
6063:
6059:
6053:
6049:
6045:
6040:
6036:
6030:
6026:
6022:
6018:
6014:
6010:
6008:0-05-002192-3
6004:
6000:
5995:
5994:
5989:
5981:
5977:
5973:
5969:
5965:
5961:
5957:
5953:
5948:
5945:
5942:
5937:
5934:
5931:
5926:
5923:
5919:
5914:
5911:
5908:
5903:
5900:
5896:
5895:Herstein 1964
5891:
5888:
5884:
5879:
5876:
5873:, p. 228
5872:
5867:
5864:
5861:, p. 116
5860:
5859:Bourbaki 1998
5855:
5852:
5845:
5826:
5806:
5801:
5797:
5791:
5787:
5783:
5778:
5775:
5770:
5767:
5762:
5758:
5754:
5746:
5742:
5724:
5720:
5697:
5693:
5672:
5667:
5663:
5659:
5639:
5634:
5630:
5626:
5603:
5577:
5571:
5551:
5545:
5539:
5536:
5533:
5527:
5522:
5518:
5514:
5505:
5500:
5496:
5492:
5445:
5425:
5422:
5412:
5408:
5385:
5365:
5362:
5352:
5348:
5325:
5322:
5317:
5313:
5302:
5298:
5292:
5272:
5269:
5264:
5260:
5249:
5245:
5239:
5219:
5216:
5211:
5207:
5196:
5192:
5186:
5181:
5177:
5166:
5162:
5156:
5136:
5133:
5130:
5125:
5121:
5117:
5114:
5111:
5106:
5102:
5090:
5086:
5080:
5075:
5071:
5067:
5059:
5017:
5013:
5009:
5004:
5000:
4979:
4976:
4971:
4967:
4963:
4958:
4954:
4950:
4945:
4941:
4937:
4932:
4928:
4924:
4916:
4898:
4894:
4885:
4881:
4877:
4871:
4868:
4861:
4857:
4854:
4851:
4848:
4847:
4843:
4841:
4839:
4834:
4830:
4826:
4822:
4818:
4813:
4811:
4807:
4799:
4797:
4795:
4794:affine scheme
4792:
4788:
4783:
4781:
4777:
4773:
4769:
4764:
4761:
4757:
4753:
4747:
4740:
4735:
4727:
4725:
4723:
4707:
4695:
4669:
4665:
4660:
4656:
4651:
4645:
4640:
4636:
4632:
4628:
4624:
4620:
4616:
4613:
4607:
4599:
4594:
4589:
4585:
4580:
4575:
4571:
4567:
4563:
4560:
4556:
4553:
4552:localizations
4549:
4545:
4541:
4536:
4531:
4527:
4522:
4519:
4514:
4510:
4504:
4500:
4494:
4490:
4484:
4480:
4475:
4471:
4467:
4463:
4460:
4456:
4452:
4448:
4444:
4440:
4437:
4433:
4429:
4426:
4425:quotient ring
4422:
4418:
4414:
4410:
4406:
4403:
4399:
4395:
4394:
4390:
4388:
4386:
4382:
4365:
4360:
4357:
4352:
4338:
4333:
4331:
4314:
4308:
4305:
4300:
4297:
4293:
4288:
4282:
4279:
4274:
4271:
4267:
4246:
4243:
4238:
4234:
4230:
4227:
4222:
4218:
4196:
4191:
4188:
4183:
4169:
4164:
4150:
4137:
4136:prime numbers
4132:
4130:
4126:
4122:
4118:
4114:
4110:
4106:
4102:
4098:
4094:
4091:if, whenever
4090:
4089:
4088:prime element
4084:
4079:
4077:
4076:
4070:
4068:
4065:
4060:
4056:
4051:
4047:
4043:
4039:
4035:
4031:
4027:
4023:
4019:
4015:
4010:
4008:
4004:
4000:
3999:
3993:
3990:
3986:
3981:
3977:
3973:
3969:
3965:
3961:
3957:
3956:
3951:
3947:
3944:
3941:
3937:
3933:
3929:
3924:
3922:
3916:
3908:
3903:
3902:fiber product
3879:
3874:
3871:
3868:
3865:
3856:
3853:
3850:
3828:
3824:
3820:
3817:
3814:
3809:
3805:
3801:
3798:
3789:
3786:
3783:
3747:
3730:
3692:
3666:
3663:
3658:
3654:
3648:
3644:
3620:
3617:
3614:
3605:
3602:
3596:
3590:
3587:
3584:
3575:
3572:
3566:
3561:
3557:
3533:
3530:
3527:
3518:
3515:
3509:
3503:
3500:
3497:
3488:
3485:
3479:
3474:
3470:
3461:
3432:
3419:
3415:
3414:
3398:
3395:
3375:
3355:
3347:
3346:
3321:
3317:
3314:
3308:
3305:
3298:
3294:
3291:
3286:
3283:
3280:
3277:
3269:
3262:
3259:
3253:
3250:
3246:
3242:
3239:
3234:
3228:
3223:
3217:
3211:
3199:
3195:
3192:
3186:
3183:
3176:
3172:
3169:
3164:
3156:
3149:
3146:
3140:
3137:
3133:
3129:
3126:
3121:
3118:
3115:
3112:
3106:
3101:
3095:
3089:
3082:
3081:
3079:
3078:unit interval
3075:
3071:
3068:
3063:
3058:
3054:
3049:
3044:
3040:
3021:
3018:
3006:
3000:
2978:
2974:
2970:
2967:
2964:
2959:
2955:
2948:
2945:
2942:
2939:
2936:
2916:
2893:
2890:
2883:
2874:
2870:
2866:
2863:
2860:
2855:
2851:
2844:
2836:
2813:
2808:
2801:
2796:
2785:
2782:
2779:
2776:
2756:
2753:
2750:
2747:
2727:
2707:
2687:
2667:
2659:
2655:
2651:
2648:
2644:
2640:
2636:
2621:
2618:
2615:
2595:
2592:
2589:
2564:
2561:
2536:
2532:
2528:
2523:
2519:
2511:
2504:
2488:
2470:
2467:
2464:
2458:
2452:
2449:
2446:
2440:
2434:
2431:
2428:
2405:
2402:
2399:
2391:
2387:
2370:
2362:
2359:
2356:
2353:
2331:
2323:
2320:
2317:
2295:
2287:
2284:
2281:
2261:
2241:
2221:
2201:
2181:
2178:
2175:
2172:
2164:
2160:
2139:
2135:
2121:
2106:
2103:
2100:
2092:
2088:
2087:
2086:
2084:
2076:
2071:
2067:
2063:
2060:
2056:
2052:
2033:
1994:
1993:complex plane
1990:
1987:
1971:
1963:
1960:
1956:
1939:
1925:
1923:
1918:
1904:
1857:
1854:
1851:
1831:
1806:
1793:
1787:
1784:
1779:
1775:
1767:
1760:
1744:
1741:
1722:
1719:
1716:
1707:
1704:
1701:
1695:
1692:
1687:
1683:
1674:
1652:
1649:
1646:
1637:
1634:
1631:
1625:
1622:
1617:
1613:
1605:
1598:
1595:
1592:
1576:
1574:coefficients.
1573:
1569:
1548:
1544:
1540:
1537:
1534:
1529:
1525:
1493:
1477:
1473:
1457:
1454:
1444:
1441:
1438:
1434:
1430:
1420:
1416:
1412:
1409:
1406:
1398:
1395:
1383:
1382:
1358:
1357:finite fields
1354:
1350:
1346:
1345:finite fields
1342:
1338:
1312:
1308:
1305:
1279:
1278:
1274:
1269:
1265:
1261:
1257:
1253:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1223:
1219:
1215:
1212:
1208:
1205:
1201:
1197:
1194:
1193:zero divisors
1190:
1189:
1188:
1186:
1183:
1179:
1171:
1162:
1157:
1155:
1150:
1148:
1143:
1142:
1140:
1139:
1131:
1128:
1126:
1123:
1121:
1118:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1098:
1097:
1093:
1090:
1089:
1085:
1080:
1079:
1072:
1071:
1067:
1066:
1062:
1059:
1057:
1054:
1052:
1049:
1048:
1043:
1038:
1037:
1030:
1029:
1025:
1023:
1020:
1019:
1015:
1012:
1010:
1007:
1005:
1002:
1000:
997:
995:
992:
990:
987:
986:
981:
976:
975:
970:
969:
962:
959:
957:
956:Division ring
954:
952:
949:
947:
944:
942:
939:
937:
934:
932:
929:
927:
924:
922:
919:
917:
914:
913:
908:
903:
902:
897:
896:
889:
886:
884:
881:
879:
878:Abelian group
876:
875:
871:
868:
866:
863:
861:
857:
854:
852:
849:
848:
844:
839:
838:
835:
831:
826:
825:
820:
819:
814:
813:
808:
807:
802:
801:
796:
795:
790:
789:
784:
783:
778:
777:
772:
771:
766:
765:
761:
760:
759:
757:
752:
750:
746:
741:
739:
735:
730:
727:
723:
717:
713:
706:
701:
697:
693:
689:
685:
681:
677:
674:
670:
666:
654:
649:
647:
642:
640:
635:
634:
632:
631:
626:
625:
619:
615:
614:
613:
612:
611:
606:
605:
604:
599:
598:
597:
590:
586:
584:
580:
578:
574:
572:
571:Division ring
568:
567:
566:
565:
559:
554:
553:
525:
509:
507:
501:
485:
471:
470:-adic numbers
469:
464:
448:
434:
432:
427:
425:
421:
419:
412:
410:
406:
405:
404:
403:
402:
393:
389:
387:
383:
379:
375:
374:
373:
369:
367:
363:
361:
357:
355:
351:
349:
345:
343:
339:
338:
337:
333:
332:
331:
330:
324:
319:
318:
309:
305:
304:
303:
299:
295:
291:
289:
285:
284:
283:
279:
275:
271:
270:
269:
265:
264:
263:
262:
237:
233:
224:
221:
214:
213:Terminal ring
210:
187:
183:
182:
181:
177:
175:
171:
169:
165:
163:
159:
157:
153:
152:
151:
150:
149:
142:
138:
136:
132:
130:
126:
125:
124:
123:
122:
115:
111:
109:
105:
103:
99:
95:
91:
89:
85:
84:
83:
82:Quotient ring
79:
77:
73:
71:
67:
66:
65:
64:
55:
54:
51:
46:→ Ring theory
45:
40:
35:
30:
19:
6412:
6406:
6336:
6330:
6317:
6289:
6264:
6245:
6229:. Springer.
6226:
6213:
6186:
6148:
6119:
6097:
6085:
6066:
6043:
6020:
5998:
5979:
5975:
5971:
5967:
5963:
5959:
5955:
5947:
5936:
5925:
5913:
5902:
5890:
5885:, p. 36
5878:
5866:
5854:
5744:
5740:
5057:
4914:
4883:
4879:
4875:
4870:
4832:
4828:
4820:
4816:
4814:
4810:prime number
4803:
4784:
4765:
4745:
4738:
4731:
4667:
4663:
4658:
4654:
4649:
4643:
4638:
4634:
4630:
4626:
4622:
4618:
4614:
4609:
4587:
4583:
4578:
4573:
4569:
4565:
4543:
4539:
4534:
4529:
4525:
4517:
4512:
4508:
4502:
4498:
4492:
4488:
4482:
4478:
4473:
4469:
4465:
4454:
4450:
4442:
4431:
4427:
4420:
4412:
4408:
4401:
4397:
4334:
4165:
4138:in the ring
4133:
4124:
4116:
4112:
4108:
4104:
4100:
4096:
4092:
4086:
4082:
4080:
4073:
4071:
4066:
4058:
4054:
4049:
4045:
4041:
4037:
4033:
4029:
4025:
4021:
4017:
4013:
4011:
4006:
4002:
3996:
3994:
3988:
3984:
3979:
3975:
3971:
3967:
3963:
3959:
3953:
3949:
3945:
3942:
3939:
3935:
3931:
3927:
3925:
3920:
3918:
3679:, and hence
3072:The ring of
3061:
3047:
3004:
2998:
2657:
2654:nonzero ring
2646:
2642:
2158:
2082:
2080:
2077:Non-examples
1957:The ring of
1921:
1919:The ring of
1567:
1337:real numbers
1248:
1244:
1236:
1232:
1228:
1177:
1175:
1130:Hopf algebra
1068:
1061:Vector space
1026:
966:
945:
895:Group theory
893:
858: /
822:
816:
810:
804:
798:
792:
786:
781:
780:
774:
768:
762:
753:
748:
742:
731:
725:
721:
715:
711:
704:
695:
692:divisibility
668:
662:
622:
608:
607:
603:Free algebra
601:
600:
594:
593:
562:
505:
467:
430:
399:
398:
378:Finite field
335:
327:
274:Finite field
260:
259:
186:Initial ring
146:
145:
119:
118:
61:
29:
6491:Ring theory
6145:Lang, Serge
6116:Lang, Serge
5952:Durbin 1993
5941:Nagata 1958
4823:, then the
4752:irreducible
4537:is zero or
4436:prime ideal
4423:, then the
4129:prime ideal
3773:defined by
3460:idempotents
3039:prime ideal
1989:open subset
1476:polynomials
1243:. Elements
1211:cancellable
1204:prime ideal
1115:Lie algebra
1100:Associative
1004:Total order
994:Semilattice
968:Ring theory
749:entire ring
665:mathematics
583:Simple ring
294:Jordan ring
168:Graded ring
50:Ring theory
6480:Categories
6445:0089.26501
6387:0084.26504
6269:A K Peters
6138:0848.13001
5990:References
5964:associates
4917:. Suppose
4760:nilradical
4722:isomorphic
4391:Properties
4330:GCD domain
4042:associates
3982:such that
3962:, or that
3948:, or that
3913:See also:
2418:, one has
1256:isomorphic
1200:zero ideal
1172:Definition
589:Commutator
348:GCD domain
6453:Q56049883
6429:0002-9327
6395:Q24655880
6361:0027-8424
5918:Lang 1993
5846:Citations
5827:◻
5798:⊗
5784:
5779:→
5759:⊗
5660:∑
5627:∑
5540:
5534:⊂
5515:∑
5493:∑
5418:¯
5358:¯
5308:¯
5293:∑
5255:¯
5240:∑
5202:¯
5187:∑
5172:¯
5157:∑
5134:≃
5122:⊗
5103:⊗
5084:→
5072:⊗
4964:⊗
4951:∑
4938:⊗
4925:∑
4838:injective
4736:(that is
4358:−
4306:−
4301:−
4280:−
4189:−
4062:for some
3883:¯
3860:↦
3854:⊗
3821:⋅
3802:⋅
3796:↦
3748:⊗
3739:→
3731:×
3693:⊗
3618:⊗
3588:⊗
3531:⊗
3510:−
3501:⊗
3433:⊗
3295:∈
3284:−
3243:∈
3173:∈
3130:∈
3116:−
3037:is not a
2968:…
2946:∈
2864:…
2652:over any
2619:−
2565:∈
2529:−
2441:⋅
2403:×
2360:≡
2091:zero ring
2059:manifolds
1986:connected
1794:≅
1785:−
1745:The ring
1720:−
1705:−
1693:−
1650:−
1635:−
1623:−
1538:…
1474:Rings of
1458:⋯
1455:⊃
1431:⊃
1413:⊃
1410:⋯
1407:⊃
1396:⊃
1351:, finite
1241:injective
1202:{0} is a
1125:Bialgebra
931:Near-ring
888:Lie group
856:Semigroup
678:in which
530:∞
308:Semifield
6449:Wikidata
6391:Wikidata
6379:16590434
6316:(1966),
6185:(1967).
6147:(2002).
6118:(1993),
6096:(2013).
6019:(1998).
5458:. Since
5438:for all
5378:for all
4844:See also
4791:integral
4787:spectrum
4743:implies
4111:divides
4103:divides
4024:divides
4016:divides
3968:multiple
3348:Neither
3051:form an
2660:≥ 2. If
2650:matrices
2554:for any
2321:≢
2285:≢
2274:). Then
1341:Artinian
1304:integers
1275:Examples
961:Lie ring
926:Semiring
719:implies
698:has the
688:integers
302:Semiring
288:Lie ring
70:Subrings
6437:2372791
6341:Bibcode
6318:Algebra
6205:0214415
6187:Algebra
6171:1878556
6149:Algebra
6120:Algebra
4886:-basis
4734:reduced
4576:, then
4383:. See
4099:, then
4028:, then
3955:divisor
3943:divides
3076:on the
2740:, then
2390:product
1991:of the
1572:complex
1353:domains
1335:of all
1302:of all
1260:subring
1249:regular
1182:nonzero
1092:Algebra
1084:Algebra
989:Lattice
980:Lattice
673:nonzero
504:Prüfer
106:•
6451:
6443:
6435:
6427:
6393:
6385:
6377:
6370:222624
6367:
6359:
6302:
6275:
6252:
6233:
6203:
6193:
6169:
6159:
6136:
6126:
6104:
6073:
6054:
6031:
6005:
4789:is an
4774:of an
4750:) and
4472:, and
4415:is an
4381:ideals
4335:While
2346:, but
1738:is an
1309:Every
1222:closed
1218:monoid
1120:Graded
1051:Module
1042:Module
941:Domain
860:Monoid
818:fields
738:domain
156:Module
129:Kernel
6433:JSTOR
6048:Wiley
4862:Notes
4629:with
4496:then
4459:field
4457:is a
4449:over
4434:is a
4417:ideal
4170:ring
4085:is a
3998:units
3966:is a
3952:is a
3043:zeros
2656:when
2161:is a
2157:when
1984:is a
1844:. If
1311:field
1264:field
1262:of a
1258:to a
1180:is a
1086:-like
1044:-like
982:-like
951:Field
909:-like
883:Magma
851:Group
845:-like
843:Group
770:rings
671:is a
667:, an
508:-ring
372:Field
268:Field
76:Ideal
63:Rings
6425:ISSN
6375:PMID
6357:ISSN
6300:ISBN
6273:ISBN
6250:ISBN
6231:ISBN
6191:ISBN
6157:ISBN
6124:ISBN
6102:ISBN
6071:ISBN
6052:ISBN
6029:ISBN
6003:ISBN
5974:and
5958:and
4804:The
4641:and
4633:and
4610:The
4486:and
4441:Let
4064:unit
4048:and
4036:are
4032:and
4020:and
3995:The
3930:and
3549:and
3416:The
3368:nor
3002:and
2680:and
2639:ring
2637:The
2608:and
2310:and
2214:and
2089:The
1855:>
1355:are
916:Ring
907:Ring
764:rngs
745:Lang
684:ring
6441:Zbl
6417:doi
6383:Zbl
6365:PMC
6349:doi
6134:Zbl
5966:if
5776:lim
5652:or
5537:Jac
5398:or
5285:or
5056:of
4913:of
4836:is
4815:If
4748:= 0
4741:= 0
4646:≠ 0
4637:in
4564:If
4557:An
4428:R/P
4419:in
4407:If
4131:.
4107:or
4072:An
4040:or
4012:If
4001:of
3978:in
3970:of
3958:of
3934:of
3411:is.
3045:of
2641:of
2367:mod
2328:mod
2292:mod
2254:or
2083:not
2070:UFD
1964:If
1239:is
1176:An
921:Rng
779:⊃
707:≠ 0
686:of
663:In
6482::
6447:.
6439:.
6431:.
6423:.
6413:80
6411:.
6389:.
6381:.
6373:.
6363:.
6355:.
6347:.
6337:45
6335:.
6329:.
6298:.
6294:.
6271:.
6267:.
6216:,
6201:MR
6199:.
6181:;
6167:MR
6165:.
6155:.
6132:,
6050:.
6027:.
5982:."
5978:|
5970:|
4840:.
4831:↦
4812:.
4796:.
4782:.
4657:→
4568:,
4542:=
4530:xJ
4528:=
4526:xI
4513:ax
4511:↦
4501:=
4493:ac
4491:=
4489:ab
4481:≠
4468:,
4387:.
4097:ab
4069:.
4059:ub
4057:=
3992:.
3987:=
3985:ax
3462:,
3062:fg
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