On fully idempotent rings
Bull. Korean Math. Soc. 2010 Vol. 47, No. 4, 715-726
https://doi.org/10.4134/BKMS.2010.47.4.715
Published online July 1, 2010
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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