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.

For a ring R with Jacobson radical J, the nonnegative powers Jⁿ are defined by using the product of ideals.

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:

Sources

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Wikipedia