In mathematics, the Köthe conjecture is a problem in ring theory, open as of 2010[update]. 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,[1] but a general solution remains elusive.
An equivalent is that the sum of two left nil ideals is a nil ideal. Kegel (1964) asked whether the sum of two nil subrings is also nil. A counterexample to this was found by Kelarev (1993).[2] It is known that the original conjecture is equivalent to the statement that sum of a nilpotent subring and a nil subring is always nil.[3]