Köthe conjecture

The Köthe conjecture asserts that the sum of two nil left ideals in any ring is nil. Equivalently, every nil one-sided ideal is contained in a nil two-sided ideal.

Connect

Nil ideal: An ideal whose every element is nilpotent, though a single exponent need not work for all elements. The conjecture asks whether sums of nil left ideals retain this elementwise nilpotence.
Nil ring: A ring in which every element is nilpotent. The conjecture can be viewed as asking whether certain sums preserve nil behavior inside rings.
Gottfried Köthe: A German mathematician whose work in ring theory gave rise to the Köthe conjecture. The conjecture bears his name and arose from questions about nil ideals.
Nilpotent ring: A ring in which every sufficiently long product of elements is zero. The conjecture concerns nil ideals, a condition weaker than nilpotence.
Nilpotent ideal: An ideal whose sufficiently long products all vanish, a stronger condition than every element being nilpotent. Confusing nil with nilpotent obscures the conjecture’s weaker, elementwise condition.
Nilpotence: The property that some positive power of an element or structure is zero. Each element of the proposed sum must have a vanishing power, without a shared bound.
Jacobson's conjecture: A conjecture asserting that the intersection of the powers of the Jacobson radical of a ring is zero. It is a separate radical-theoretic problem often discussed alongside Köthe’s conjecture.
Locally nilpotent ring: A ring in which every finitely generated subring is nilpotent. Local nilpotence is stronger than elementwise nilness and is not guaranteed by the conjecture.
Left ideal: An additive subgroup of a ring closed under multiplication by ring elements on the left. The conjecture begins with nil ideals that need not be closed under right multiplication.
Amitsur's theorem: A theorem stating that the Jacobson radical of a polynomial ring over a ring is a nil ideal. It connects the conjecture to nilness inside radicals of polynomial extensions.