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| In [[abstract algebra]], '''Jacobson's conjecture''' is an open problem in [[ring theory]] concerning the intersection of powers of the [[Jacobson radical]] of a [[Noetherian ring]].
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| It has only been proven for special types of Noetherian rings, so far. Examples exist to show that the conjecture can fail when the ring is not Noetherian on a side, so it is absolutely necessary for the ring to be two-sided Noetherian.
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| The conjecture is named for the algebraist [[Nathan Jacobson]] who posed the first version of the conjecture.
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| ==Statement==
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| For a ring ''R'' with Jacobson radical ''J'', the nonnegative powers ''J''<sup>''n''</sup> are defined by using the [[product of ideals]].
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| :''Jacobson's conjecture:'' In a right-and-left [[Noetherian ring]], <math>\bigcap_{n\in \mathbb{N}}J^n=\{0\}.</math> | |
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| In other words: "The only element of a Noetherian ring in all powers of ''J'' is 0."
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| The original conjecture posed by Jacobson in 1956<ref>{{citation
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| | last = Jacobson | first = Nathan
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| | location = 190 Hope Street, Prov., R. I.
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| | mr = 0081264
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| | page = 200
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| | publisher = American Mathematical Society
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| | series = American Mathematical Society, Colloquium Publications, vol. 37
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| | title = Structure of rings
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| | year = 1956}}. As cited by {{citation
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| | last1 = Brown | first1 = K. A.
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| | last2 = Lenagan | first2 = T. H.
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| | doi = 10.1017/S0017089500004729
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| | issue = 1
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| | journal = Glasgow Mathematical Journal
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| | mr = 641612
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| | pages = 7–8
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| | title = A note on Jacobson's conjecture for right Noetherian rings
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| | volume = 23
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| | year = 1982}}.</ref> asked about noncommutative one-sided Noetherian rings, however [[Herstein]] produced a counterexample in 1965{{sfn|Herstein|1965}} and soon after Jategaonkar produced a different example which was a left [[principal ideal domain]].{{sfn|Jategaonkar|1968}} From that point on, the conjecture was reformulated to require two-sided Noetherian rings.
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| ==Partial results==
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| Jacobson's conjecture has been verified for particular types of Noetherian rings:
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| * [[commutative ring|Commutative]] Noetherian rings all satisfy Jacobson's conjecture. This is a consequence of the [[Krull intersection theorem]].
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| * [[Fully bounded Noetherian ring]]s{{sfn|Cauchon|1974}}{{sfn|Jategaonkar|1974}}
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| * Noetherian rings with [[Krull dimension]] 1{{sfn|Lenagan|1977}}
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| * Noetherian rings satisfying the [[second layer condition]]{{sfn|Jategaonkar|1982}}
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| ==References==
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| {{Reflist}}
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| *{{citation |last=Cauchon|first=Gérard |title=Sur l'intersection des puissances du radical d'un T-anneau noethérien |language=French |journal=C. R. Acad. Sci. Paris Sér. A |volume=279 |year=1974 |pages=91–93 |mr=0347894}}
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| *{{citation |last1=Goodearl|first1=K. R. |last2=Warfield|first2=R. B., Jr. |title=An introduction to noncommutative Noetherian rings |series=London Mathematical Society Student Texts |volume=61 |edition=2 |publisher=Cambridge University Press |place=Cambridge |year=2004 |pages=xxiv+344 |isbn=0-521-83687-5 |isbn=0-521-54537-4 |mr=2080008 }}
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| *{{citation |last=Herstein |first=I. N. |title=A counterexample in Noetherian rings |journal=Proc. Nat. Acad. Sci. U.S.A. |volume=54 |year=1965 |pages=1036–1037 |issn=0027-8424 |mr=0188253}}
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| *{{citation |last=Jategaonkar |first=Arun Vinayak |title=Left principal ideal domains |journal=J. Algebra |volume=8 |year=1968 |pages=148–155 |issn=0021-8693 |mr=0218387}}
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| *{{citation|last=Jategaonkar |first=Arun Vinayak |title=Jacobson's conjecture and modules over fully bounded Noetherian rings |journal=J. Algebra |volume=30 |year=1974 |pages=103–121 |issn=0021-8693 |mr=0352170}}
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| *{{citation |last=Jategaonkar|first=Arun Vinayak |title=Solvable Lie algebras, polycyclic-by-finite groups and bimodule Krull dimension |journal=Comm. Algebra|volume=10 |year=1982 |number=1 |pages=19–69 |issn=0092-7872 |mr=674687 |doi=10.1080/00927878208822700}}
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| *{{citation |last=Lenagan|first=T. H. |title=Noetherian rings with Krull dimension one |journal=J. London Math. Soc. (2) |volume=15 |year=1977 |number=1 |pages=41–47 |issn=0024-6107 |mr=0442008}}
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| *{{citation |last=Rowen |first=Louis H. |title=Ring theory. Vol. I |series=Pure and Applied Mathematics |volume=127 |publisher=Academic Press Inc. |place=Boston, MA |year=1988 |pages=xxiv+538 |isbn=0-12-599841-4 |mr=940245}}
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| [[Category:Conjectures| ]]
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| [[Category:Ring theory]]
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