Köthe conjecture
In mathematics, the Köthe conjecture is a problem in ring theory, open . It is formulated in various ways. Suppose that R is a ring. One way to state the conjecture is that if R has no nil ideal, other than {0}, then it has no nil one-sided ideal, other than {0}. This question was posed in 1930 by Gottfried Köthe (1905–1989). The Köthe conjecture has been shown to be true for various classes of rings, such as polynomial identity rings and right Noetherian rings, but a general solution remains elusive. Equivalent formulations The conjecture has several different formulations: (Köthe conjecture) In any ring, the sum of two nil left ideals is nil. In any ring, the sum of two one-sided nil ideals is nil. In any ring, every nil left or right ideal of the ring is contained in the upper nil radical of the ring. For any ring R and for any nil ideal J of R, the matrix ideal Mn(J) is a nil ideal of Mn(R) for every n. For any ring R and for any nil ideal J of R, the matrix ideal M2(J) is a nil ideal of M2(R). For any ring R, the upper