Köthe conjecture

The conjecture has several different formulations:Krempa, J., "Logical connections between some open problems concerning nil rings," Fundamenta Mathematicae 76 (1972), no. 2, 121–130.Lam, T.Y., A First Course in Noncommutative Rings (2001), p.171.Lam, T.Y., Exercises in Classical Ring Theory (2003), p. 160.

1. (Köthe conjecture) In any ring, the sum of two nil left ideals is nil.
2. In any ring, the sum of two one-sided nil ideals is nil.
3. In any ring, every nil left or right ideal of the ring is contained in the upper nil radical of the ring.
4. For any ring R and for any nil ideal J of R, the matrix ideal M_n(J) is a nil ideal of M_n(R) for every n.
5. For any ring R and for any nil ideal J of R, the matrix ideal M₂(J) is a nil ideal of M₂(R).
6. For any ring R, the upper nilradical of M_n(R) is the set of matrices with entries from the upper nilradical of R for every positive integer n.
7. For any ring R and for any nil ideal J of R, the polynomials with indeterminate x and coefficients from J lie in the Jacobson radical of the polynomial ring R[x].
8. For any ring R, the Jacobson radical of R[x] consists of the polynomials with coefficients from the upper nilradical of R.

Necessary and sufficient conditions under which the statement 3 given above does not hold have been given recently. Pandey, S. K., "A note on upper nilradicals and one-sided nil ideals," Serdica Mathematical Journal 50 (2024), no. 3-4 .

A conjecture by Amitsur read: "If J is a nil ideal in R, then J[x] is a nil ideal of the polynomial ring R[x]."Amitsur, S. A. Nil radicals. Historical notes and some new results Rings, modules and radicals (Proc. Internat. Colloq., Keszthely, 1971), pp. 47–65. Colloq. Math. Soc. János Bolyai, Vol. 6, North-Holland, Amsterdam, 1973. This conjecture, if true, would have proven the Köthe conjecture through the equivalent statements above, however a counterexample was produced by Agata Smoktunowicz.Smoktunowicz, Agata. Polynomial rings over nil rings need not be nil J. Algebra 233 (2000), no. 2, p. 427–436. While not a disproof of the Köthe conjecture, this fueled suspicions that the Köthe conjecture may be false.Lam, T.Y., A First Course in Noncommutative Rings (2001), p.171.

Kegel proved that a ring which is the direct sum of two nilpotent subrings is itself nilpotent.{{CN|date=October 2023}} The question arose whether or not "nilpotent" could be replaced with "locally nilpotent" or "nil". Partial progress was made when KelarevKelarev, A. V., A sum of two locally nilpotent rings may not be nil, Arch. Math. 60 (1993), p431–435. produced an example of a ring which isn't nil, but is the direct sum of two locally nilpotent rings. This demonstrates that Kegel's question with "locally nilpotent" replacing "nilpotent" is answered in the negative.

The sum of a nilpotent subring and a nil subring is always nil.Ferrero, M., Puczylowski, E. R., On rings which are sums of two subrings, Arch. Math. 53 (1989), p4–10.

• {{Citation | last1=Köthe | first1=Gottfried | title=Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollständig reduzibel ist | doi=10.1007/BF01194626 | year=1930 | journal=Mathematische Zeitschrift | volume=32 | issue=1 | pages=161–186| s2cid=123292297 }}

{{Reflist}}

• [http://planetmath.org/?op=getobj&from=objects&id=3691 PlanetMath page]
• [http://www.math.bas.bg/serdica/2001/2001-159-170.pdf Survey paper (PDF)]

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Category:Ring theory

Category:Conjectures

Category:Unsolved problems in mathematics