Köthe conjecture
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In mathematics, the Köthe conjecture is a problem in ring theory, open as of 2005. 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)
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). 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.
[edit] References
- Gottfried Köthe, Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollständig reduzibel ist, Math. Zeitschrift, 32 (1930), 161-186.
