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 asked about
noncommutative one-sided Noetherian rings, however
Israel Nathan Herstein produced a
counterexample in 1965, and soon afterwards, Arun Vinayak 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. ==Partial results==