63:
22:
165:
322:
a polynomial function of degree 1, so an element has a factorization only if it has finitely many zeroes. In the case of the algebraic integers there are no irreducible elements at all, since for any algebraic integer its square root (for instance) is also an algebraic integer. This shows in
808:
of them: this is equivalent to the statement that an ideal which is generated by two elements is also generated by a single element, and induction demonstrates that all finitely generated ideals are principal. The expression of the greatest common divisor of two elements of a PID as a linear
887:
is not
Noetherian, then there exists an infinite ascending chain of finitely generated ideals, so in a Bézout domain an infinite ascending chain of principal ideals. (4) and (2) are thus equivalent.
883:
The equivalence of (1) and (2) was noted above. Since a Bézout domain is a GCD domain, it follows immediately that (3), (4) and (5) are equivalent. Finally, if
816:
Note that the above gcd condition is stronger than the mere existence of a gcd. An integral domain where a gcd exists for any two elements is called a
865:
928:
is principal, a local ring is a Bézout domain iff it is a valuation domain. Moreover, a valuation domain with noncyclic (equivalently non-
901:
Consequently, one may view the equivalence "Bézout domain iff Prüfer domain and GCD-domain" as analogous to the more familiar "PID iff
127:
99:
106:
225:
207:
146:
49:
189:
113:
80:
35:
285:. The theory of Bézout domains retains many of the properties of PIDs, without requiring the Noetherian property.
924:. So the localization of a Bézout domain at a prime ideal is a valuation domain. Since an invertible ideal in a
95:
84:
1161:
1178:
894:, i.e., a domain in which each finitely generated ideal is invertible, or said another way, a commutative
1156:
939:
is the value group of some valuation domain. This gives many examples of non-Noetherian Bézout domains.
855:
278:
1051:
929:
329:
are Bézout domains. Any non-Noetherian valuation ring is an example of a non-noetherian Bézout domain.
259:
180:
946:
are domains whose finitely generated right ideals are principal right ideals, that is, of the form
800:
A ring is a Bézout domain if and only if it is an integral domain in which any two elements have a
1151:
801:
73:
966:
commutative domain is an Ore domain. Right Bézout domains are also right semihereditary rings.
810:
255:
120:
909:
270:
41:
1183:
1133:
1114:
1078:
8:
821:
295:
805:
355:
319:
396:, and the ideal generated by all these quotients of is not finitely generated (and so
1102:
315:
175:
511:
be the minimal exponent such that at least one of them has a nonzero coefficient of
1140:
1094:
1066:
703:; we shall show it is in fact a greatest common divisor by showing that it lies in
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891:
263:
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1110:
1074:
902:
895:
388:
with zero constant term can be divided indefinitely by noninvertible elements of
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274:
251:
247:
974:
Some facts about modules over a PID extend to modules over a Bézout domain. Let
921:
326:
1070:
1005:
1172:
1106:
936:
917:
876:
825:
292:
289:
820:
and thus Bézout domains are GCD domains. In particular, in a Bézout domain,
1121:
318:. In case of entire functions, the only irreducible elements are functions
933:
913:
239:
959:
925:
817:
314:(functions holomorphic on the whole complex plane) and the ring of all
282:
185:
mathematical proofs are used in place of citations to existing sources.
1085:
Helmer, Olaf (1940), "Divisibility properties of integral functions",
277:, so it could have non-finitely generated ideals; if so, it is not a
62:
828:(but as the algebraic integer example shows, they need not exist).
323:
both cases that the ring is not a UFD, and so certainly not a PID.
999:
310:
Examples of Bézout domains that are not PIDs include the ring of
558:
are not relatively prime, there is a greatest common divisor of
962:. This fact is not interesting in the commutative case, since
958:. One notable result is that a right Bézout domain is a right
908:
Prüfer domains can be characterized as integral domains whose
566:
in this UFD that has constant term 1, and therefore lies in
332:
The following general construction produces a Bézout domain
273:(PID) is a Bézout domain, but a Bézout domain need not be a
1128:, Boston, Mass.: Allyn and Bacon Inc., pp. x+180,
340:
that is not a field, for instance from a PID; the case
1002:(a commutative semifir is precisely a Bézout domain.)
687:, is divisible by any nonzero constant, the constant
507:
nonzero; if both have a zero constant term, then let
384:. This ring is not Noetherian, since an element like
87:. Unsourced material may be challenged and removed.
767:has a zero constant term, and so is a multiple in
1120:
258:holds for every pair of elements, and that every
1170:
875:factors into a product of irreducibles (R is an
661:. Since any element without constant term, like
969:
835:, the following conditions are all equivalent:
866:ascending chain condition on principal ideals
1084:
721:respectively by the Bézout coefficients for
1049:
990:is flat if and only if it is torsion-free.
932:) value group is not Noetherian, and every
262:is principal. Bézout domains are a form of
254:is also a principal ideal. This means that
50:Learn how and when to remove these messages
350:is the basic example to have in mind. Let
415:It suffices to prove that for every pair
336:that is not a UFD from any Bézout domain
226:Learn how and when to remove this message
208:Learn how and when to remove this message
147:Learn how and when to remove this message
1139:
1033:
789:does as well, which completes the proof.
538:We may therefore assume at least one of
1171:
158:
85:adding citations to reliable sources
56:
15:
392:, which are still noninvertible in
288:Bézout domains are named after the
13:
573:We may therefore also assume that
14:
1195:
1052:"Bezout rings and their subrings"
31:This article has multiple issues.
546:has a nonzero constant term. If
472:, it suffices to prove this for
376:, the subring of polynomials in
163:
61:
20:
1043:
1036:, Ch I, §2, no 4, Proposition 3
982:finitely generated module over
595:, and some constant polynomial
570:; we can divide by this factor.
72:needs additional citations for
39:or discuss these issues on the
1027:
1018:
499:We may assume the polynomials
1:
1099:10.1215/s0012-7094-40-00626-3
1011:
795:
404:). One shows as follows that
1059:Proc. Cambridge Philos. Soc.
970:Modules over a Bézout domain
842:is a principal ideal domain.
809:combination is often called
617:is a Bézout domain, the gcd
307:All PIDs are Bézout domains.
7:
1157:Encyclopedia of Mathematics
993:
942:In noncommutative algebra,
856:unique factorization domain
771:of the constant polynomial
301:
279:unique factorization domain
183:. The specific problem is:
10:
1200:
813:, whence the terminology.
1071:10.1017/s0305004100042791
871:Every nonzero nonunit in
775:, and therefore lies in
581:are relatively prime in
400:has no factorization in
260:finitely generated ideal
250:in which the sum of two
978:be a Bézout domain and
802:greatest common divisor
691:is a common divisor in
527:is a common divisor of
916:(equivalently, at all
625:of the constant terms
554:viewed as elements of
468:have a common divisor
380:with constant term in
281:(UFD), but is still a
271:principal ideal domain
890:A Bézout domain is a
1050:Cohn, P. M. (1968),
944:right Bézout domains
831:For a Bézout domain
585:, so that 1 lies in
190:improve this article
179:to meet Knowledge's
81:improve this article
1179:Commutative algebra
1145:Commutative algebra
753:with constant term
739:gives a polynomial
408:is a Bézout domain.
806:linear combination
356:field of fractions
316:algebraic integers
1141:Bourbaki, Nicolas
1126:Commutative rings
1122:Kaplansky, Irving
922:valuation domains
811:Bézout's identity
725:with respect to
535:and divide by it.
488:, since the same
256:Bézout's identity
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181:quality standards
172:This article may
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252:principal ideals
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934:totally ordered
903:Dedekind domain
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515:; one can find
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248:integral domain
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1093:(2): 345–356,
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896:semihereditary
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864:satisfies the
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848:is Noetherian.
843:
797:
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729:
713:. Multiplying
683:
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644:
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613:. Also, since
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296:Étienne Bézout
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1152:"Bezout ring"
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1087:Duke Math. J.
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1034:Bourbaki 1989
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937:abelian group
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920:) ideals are
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910:localizations
906:
904:
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893:
892:Prüfer domain
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877:atomic domain
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449:divides both
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293:mathematician
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98: –
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92:Find sources:
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70:This article
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1044:Bibliography
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822:irreducibles
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188:Please help
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79:Please help
74:verification
71:
47:
40:
34:
33:Please help
30:
1184:Ring theory
1006:Bézout ring
905:and UFD".
785:. But then
240:mathematics
198:August 2024
192:if you can.
137:August 2024
1173:Categories
1012:References
960:Ore domain
926:local ring
818:GCD domain
804:that is a
796:Properties
523:such that
439:such that
362:, and put
283:GCD domain
107:newspapers
36:improve it
1162:EMS Press
1107:0012-7094
950:for some
898:domain.)
42:talk page
1143:(1989),
1124:(1970),
994:See also
930:discrete
639:lies in
603:lies in
496:will do.
302:Examples
174:require
1164:, 2001
1134:0254021
1115:0001851
1079:0222065
1000:Semifir
986:. Then
918:maximal
912:at all
868:(ACCP).
757:. Then
354:be the
176:cleanup
121:scholar
1132:
1113:
1105:
1077:
858:(UFD).
290:French
246:is an
123:
116:
109:
102:
94:
1055:(PDF)
964:every
914:prime
854:is a
826:prime
128:JSTOR
114:books
1103:ISSN
1024:Cohn
824:are
732:and
717:and
699:and
632:and
577:and
562:and
550:and
531:and
503:and
480:and
464:and
453:and
269:Any
242:, a
100:news
1095:doi
1067:doi
954:in
743:in
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599:in
519:in
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