Jacobson's conjecture

Jacobson's conjecture asserts that for every right Noetherian ring, the intersection of all positive powers of its Jacobson radical is zero.

Right Noetherian ring: A ring in which every ascending chain of right ideals eventually stabilizes. Right Noetherianity is the hypothesis under which the conjecture predicts separation.
I-adic filtration: The descending sequence of powers of an ideal, used to study a ring or module through successive quotients. The Jacobson radical powers form the filtration whose intersection is at issue.
Nathan Jacobson: An American algebraist whose work shaped the modern theory of rings and noncommutative algebra. His name is attached to the radical central to the conjecture.
Jacobson radical problem: A family of questions about the structure and behavior of Jacobson radicals in rings. Jacobson’s conjecture is one specific unresolved question within radical theory.
Ideal power: For an ideal I, its nth power is the ideal generated by products of n elements of I. The conjecture concerns every positive integer power of the radical.
Krull intersection theorem: A theorem giving conditions under which the intersection of powers of an ideal in a Noetherian ring is zero. It proves a related separation result in commutative settings, unlike the conjecture’s right Noetherian setting.
Jacobson radical: The intersection of a ring’s maximal right ideals; it also equals the intersection of its maximal left ideals. Jacobson’s radical provides the ideal whose powers motivate the conjecture.
Right Artinian ring: A ring in which descending chains of right ideals eventually stabilize. For this stronger finiteness condition, the radical is nilpotent, so the intersection vanishes.
Intersection of ideals: The set of elements common to a collection of ideals, itself an ideal. The claimed zero intersection means no nonzero element lies in every radical power.
Nakayama's lemma: A theorem that forces a finitely generated module to vanish when multiplication by a suitable ideal leaves it unchanged. Radical powers interact with finite generation through Nakayama-type arguments.

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Köthe conjectureRelated: It is a separate radical-theoretic problem often discussed alongside Köthe’s conjecture.