Jacobson's conjecture

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.

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.

The conjecture is named for the algebraist Nathan Jacobson who posed the first version of the conjecture.

Statement

For a ring R with Jacobson radical J, the nonnegative powers $J^{n}$ are defined by using the product of ideals.

Jacobson's conjecture: In a right-and-left Noetherian ring,

\bigcap _{n\in \mathbb {N} }J^{n}=\{0\}.

In other words: "The only element of a Noetherian ring in all powers of J is 0."

The original conjecture posed by Jacobson in 1956^[1] asked about noncommutative one-sided Noetherian rings, however Israel Nathan Herstein produced a counterexample in 1965,^[2] and soon afterwards, Arun Vinayak Jategaonkar produced a different example which was a left principal ideal domain.^[3] From that point on, the conjecture was reformulated to require two-sided Noetherian rings.

Partial results

Jacobson's conjecture has been verified for particular types of Noetherian rings:

Commutative Noetherian rings all satisfy Jacobson's conjecture. This is a consequence of the Krull intersection theorem.
Fully bounded Noetherian rings^[4]^[5]
Noetherian rings with Krull dimension 1^[6]
Noetherian rings satisfying the second layer condition^[7]

References

^ Jacobson, Nathan (1956), Structure of rings, American Mathematical Society, Colloquium Publications, vol. 37, 190 Hope Street, Providence, R. I.: American Mathematical Society, p. 200, MR 0081264{{citation}}: CS1 maint: location (link). As cited by Brown, K. A.; Lenagan, T. H. (1982), "A note on Jacobson's conjecture for right Noetherian rings", Glasgow Mathematical Journal, 23 (1): 7–8, doi:10.1017/S0017089500004729, MR 0641612.
^ Herstein 1965.
^ Jategaonkar 1968.
^ Cauchon 1974.
^ Jategaonkar 1974.
^ Lenagan 1977.
^ Jategaonkar 1982.

Sources

Cauchon, Gérard (1974), "Sur l'intersection des puissances du radical d'un T-anneau noethérien", Comptes Rendus de l'Académie des Sciences, Série A (in French), 279: 91–93, MR 0347894
Goodearl, K. R.; Warfield, R. B. Jr. (2004), An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts, vol. 61 (2 ed.), Cambridge: Cambridge University Press, pp. xxiv+344, ISBN 0-521-54537-4, MR 2080008
Herstein, I. N. (1965), "A counterexample in Noetherian rings", Proceedings of the National Academy of Sciences of the United States of America, 54 (4): 1036–1037, Bibcode:1965PNAS...54.1036H, doi:10.1073/pnas.54.4.1036, ISSN 0027-8424, MR 0188253, PMC 219788, PMID 16578617
Jategaonkar, Arun Vinayak (1968), "Left principal ideal domains", J. Algebra, 8 (2): 148–155, doi:10.1016/0021-8693(68)90040-9, ISSN 0021-8693, MR 0218387
Jategaonkar, Arun Vinayak (1974), "Jacobson's conjecture and modules over fully bounded Noetherian rings", J. Algebra, 30 (1–3): 103–121, doi:10.1016/0021-8693(74)90195-1, ISSN 0021-8693, MR 0352170
Jategaonkar, Arun Vinayak (1982), "Solvable Lie algebras, polycyclic-by-finite groups and bimodule Krull dimension", Comm. Algebra, 10 (1): 19–69, doi:10.1080/00927878208822700, ISSN 0092-7872, MR 0674687
Lenagan, T. H. (1977), "Noetherian rings with Krull dimension one", J. London Math. Soc., Series 2, 15 (1): 41–47, doi:10.1112/jlms/s2-15.1.41, ISSN 0024-6107, MR 0442008
Rowen, Louis H. (1988), Ring theory. Vol. I, Pure and Applied Mathematics, vol. 127, Boston, MA: Academic Press Inc., pp. xxiv+538, ISBN 0-12-599841-4, MR 0940245

[1] Jacobson, Nathan (1956), Structure of rings, American Mathematical Society, Colloquium Publications, vol. 37, 190 Hope Street, Providence, R. I.: American Mathematical Society, p. 200, MR 0081264{{citation}}: CS1 maint: location (link). As cited by Brown, K. A.; Lenagan, T. H. (1982), "A note on Jacobson's conjecture for right Noetherian rings", Glasgow Mathematical Journal, 23 (1): 7–8, doi:10.1017/S0017089500004729, MR 0641612.

[FOOTNOTEHerstein1965-2] Herstein 1965.

[FOOTNOTEJategaonkar1968-3] Jategaonkar 1968.

[FOOTNOTECauchon1974-4] Cauchon 1974.

[FOOTNOTEJategaonkar1974-5] Jategaonkar 1974.

[FOOTNOTELenagan1977-6] Lenagan 1977.

[FOOTNOTEJategaonkar1982-7] Jategaonkar 1982.

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