Rahul Sharma (Editor)

Jacobson's conjecture

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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.

Contents

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 Jn are defined by using the product of ideals.

Jacobson's conjecture: In a right-and-left Noetherian ring, ⋂ n ∈ 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 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.

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
  • Noetherian rings with Krull dimension 1
  • Noetherian rings satisfying the second layer condition

    • References

    Jacobson's conjecture Wikipedia
    (Text) CC BY-SA