Köthe conjecture

Köthe conjecture

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.

References

*Gottfried Köthe, "Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollständig reduzibel ist", Math. Zeitschrift, 32 (1930), 161-186.

External links

* [http://planetmath.org/?op=getobj&from=objects&id=3691 PlanetMath page]


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