Rings Whose Nilpotent Elements form a Levitzki Radical Ring

Abstract

It is shown that a locally 2-primal ring, but not 2-primal, can be always constructed from any given a 2-primal ring. Locally 2-primal rings are NI but we show that there are NI rings which are not locally 2-primal. We prove that every minimal noncommutative (locally) 2-primal ring is isomorphic to the 2 by 2 upper triangular matrix ring over GF(2). By Smoktunowicz (2000), a nil ring R need not be locally 2-primal, but we show that it is the case if and only if R is a Levitzki radical ring. We also prove that the local 2-primalness is inherited by polynomial rings, but not by power series rings. However in the case of rings of bounded index of nilpotency, a ring is locally 2-primal if and only if so is its power series ring.

Keywords:

• Levitzki radical
• (Locally) 2-primal ring
• NI ring
• Prime radical
• Upper nilradical

Mathematics Subject Classification:

• Primary 16N40
• Secondary 16S36, 16U20

ACKNOWLEDGMENTS

The authors are very grateful to the referee for various valuable comments leading to the improvement of the article. Y. Lee was supported by the Korea Research Foundation Grant (KRF-2005-015-C00011), and the N.K. Kim was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD)(KRF-2005-C00011).

Notes

Communicated by M. Ferrero.