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.
For a ring R with Jacobson radical J, the nonnegative powers Jn are defined by using the product of ideals.
- Jacobson's conjecture: In a right-and-left Noetherian ring,
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 asked about noncommutative one-sided Noetherian rings, however Herstein produced a counterexample in 1965 and soon after Jategaonkar produced a different example which was a left principal ideal domain. From that point on, the conjecture was reformulated to require two-sided Noetherian rings.
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
- Noetherian rings with Krull dimension 1
- Noetherian rings satisfying the second layer condition
- Jacobson, Nathan (1956), Structure of rings, American Mathematical Society, Colloquium Publications, vol. 37, 190 Hope Street, Prov., R. I.: American Mathematical Society, p. 200, MR 0081264. 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 641612.
- Herstein 1965.
- Jategaonkar 1968.
- Cauchon 1974.
- Jategaonkar 1974.
- Lenagan 1977.
- Jategaonkar 1982.
- Cauchon, Gérard (1974), "Sur l'intersection des puissances du radical d'un T-anneau noethérien", C. R. Acad. Sci. Paris Sér. 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 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", Proc. Nat. Acad. Sci. U.S.A. 54: 1036–1037, ISSN 0027-8424, MR 0188253
- Jategaonkar, Arun Vinayak (1968), "Left principal ideal domains", J. Algebra 8: 148–155, ISSN 0021-8693, MR 0218387
- Jategaonkar, Arun Vinayak (1974), "Jacobson's conjecture and modules over fully bounded Noetherian rings", J. Algebra 30: 103–121, 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 674687
- Lenagan, T. H. (1977), "Noetherian rings with Krull dimension one", J. London Math. Soc. (2) 15 (1): 41–47, ISSN 0024-6107, MR 0442008
- Rowen, Louis H. (1988), Ring theory. Vol. I, Pure and Applied Mathematics 127, Boston, MA: Academic Press Inc., pp. xxiv+538, ISBN 0-12-599841-4, MR 940245