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