Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2010; 47(4): 715-726

Printed July 1, 2010

https://doi.org/10.4134/BKMS.2010.47.4.715

Copyright © The Korean Mathematical Society.

On fully idempotent rings

Young Cheol Jeon, Nam Kyun Kim, and Yang Lee

Korea Science Academy, Hanbat National University, Busan National University

Abstract

We continue the study of fully idempotent rings initiated by Courter. It is shown that a (semi)prime ring, but not fully idempotent, can be always constructed from any (semi)prime ring. It is shown that the full idempotence is both Morita invariant and a hereditary radical property, obtaining $hs({\rm Mat}_n(R))={\rm Mat}_n(hs(R))$ for any ring $R$ where $hs(-)$ means the sum of all fully idempotent ideals. A non-semiprimitive fully idempotent ring with identity is constructed from the Smoktunowicz's simple nil ring. It is proved that the full idempotence is preserved by the classical quotient rings. More properties of fully idempotent rings are examined and necessary examples are found or constructed in the process.

Keywords: fully idempotent ring, (weakly) regular ring, hereditary radical, classical quotient ring

MSC numbers: 16D25, 16E50, 16N60, 16N80

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