A nil implies nilpotent theorem for left ideals. (English) Zbl 0722.16011
I. N. Herstein conjectured [in J. Math. Anal. Appl. 15, 91-96 (1966; Zbl 0139.260)] that if \(I\subset J\) are left ideals of a left Noetherian ring with some power of each element of J lying in I, then \(J^ n\subseteq I\) for some n; he also proved the conjecture when R satisfies a polynomial identity. In the present note some very pretty arguments are brought to bear to prove Herstein’s conjecture when J/I is an Artinian left R-module. As corollaries, the conjecture is verified when R is left fully bounded, or simple, or the integral group ring of a polycyclic-by-finite group.
Reviewer: K.A.Brown (Glasgow)
MSC:
16N40 | Nil and nilpotent radicals, sets, ideals, associative rings |
16P40 | Noetherian rings and modules (associative rings and algebras) |
16D25 | Ideals in associative algebras |
16S34 | Group rings |
Keywords:
fully bounded Noetherian ring; left ideals; left Noetherian ring; Herstein’s conjecture; Artinian left R-module; integral group ring; polycyclic-by-finite groupCitations:
Zbl 0139.260References:
[1] | Brookes, C. J.B, Modules over polycyclic groups, (Proc. London Math. Soc., 57 (1988)), 88-108 · Zbl 0644.20010 |
[2] | Herstein, I. N., A theorem on left Noetherian rings, J. Math. Anal. Appl., 15, 91-96 (1966) · Zbl 0139.26001 |
[3] | Herstein, I. N., A nil-nilpotent type of theorem, (Barroso, J. A., Aspects of Mathematics and Its Applications (1986), North-Holland: North-Holland Amsterdam) · Zbl 0593.16007 |
[4] | McConnell, J. C.; Robson, J. C., Noncommutative Noetherian Rings (1987), Wiley: Wiley Chichester/New York · Zbl 0644.16008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.