词条 | Köthe conjecture |
释义 |
In mathematics, the Köthe conjecture is a problem in ring theory, open {{As of|2010|lc=on}}. 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] and right Noetherian rings,[2] but a general solution remains elusive. Equivalent formulationsThe conjecture has several different formulations:[3][4][5]
Related problemsA 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]."[6] This conjecture, if true, would have proven the Köthe conjecture through the equivalent statements above, however a counterexample was produced by Agata Smoktunowicz.[7] While not a disproof of the Köthe conjecture, this fueled suspicions that the Köthe conjecture may be false in general.[8] In {{harv|Kegel|1962}}, it was proven that a ring which is the direct sum of two nilpotent subrings is itself nilpotent. The question arose whether or not "nilpotent" could be replaced with "locally nilpotent" or "nil". Partial progress was made when Kelarev[9] 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.[10] References
1. ^John C. McConnell, James Christopher Robson, Lance W. Small , Noncommutative Noetherian rings (2001), p. 484. 2. ^Lam, T.Y., A First Course in Noncommutative Rings (2001), p.164. 3. ^Krempa, J., “Logical connections between some open problems concerning nil rings,” Fundamenta Mathematicae 76 (1972), no. 2, 121–130. 4. ^Lam, T.Y., A First Course in Noncommutative Rings (2001), p.171. 5. ^Lam, T.Y., Exercises in Classical Ring Theory (2003), p. 160. 6. ^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. 7. ^Smoktunowicz, Agata. Polynomial rings over nil rings need not be nil J. Algebra 233 (2000), no. 2, p. 427–436. 8. ^Lam, T.Y., A First Course in Noncommutative Rings (2001), p.171. 9. ^Kelarev, A. V., A sum of two locally nilpotent rings may not be nil, Arch. Math. 60 (1993), p431–435. 10. ^Ferrero, M., Puczylowski, E. R., On rings which are sums of two subrings, Arch. Math. 53 (1989), p4–10. External links
2 : Ring theory|Conjectures |
随便看 |
|
开放百科全书收录14589846条英语、德语、日语等多语种百科知识,基本涵盖了大多数领域的百科知识,是一部内容自由、开放的电子版国际百科全书。