SEMICENTRAL IDEMPOTENTS IN A RING

Title & Authors
SEMICENTRAL IDEMPOTENTS IN A RING
Han, Juncheol; Lee, Yang; Park, Sangwon;

Abstract
Let R be a ring with identity 1, I(R) be the set of all nonunit idempotents in R and $\small{S_{\ell}}$(R) (resp. $\small{S_r}$(R)) be the set of all left (resp. right) semicentral idempotents in R. In this paper, the following are investigated: (1) $\small{e{\in}S_{\ell}(R)}$ (resp. $\small{e{\in}S_r(R)}$) if and only if re=ere (resp. er=ere) for all nilpotent elements $\small{r{\in}R}$ if and only if $\small{fe{\in}I(R)}$ (resp. $\small{ef{\in}I(R)}$) for all $\small{f{\in}I(R)}$ if and only if fe=efe (resp. ef=efe) for all $\small{f{\in}I(R)}$ if and only if fe=efe (resp. ef=efe) for all $\small{f{\in}I(R)}$ which are isomorphic to e if and only if $\small{(fe)^n=(efe)^n}$ (resp. $\small{(ef)^n=(efe)^n}$) for all $\small{f{\in}I(R)}$ which are isomorphic to e where n is some positive integer; (2) For a ring R having a complete set of centrally primitive idempotents, every nonzero left (resp. right) semicentral idempotent is a finite sum of orthogonal left (resp. right) semicentral primitive idempotents, and eRe has also a complete set of primitive idempotents for any $\small{0{\neq}e{\in}S_{\ell}(R)}$ (resp. 0$\small{0{\neq}e{\in}S_r(R)}$).
Keywords
left (resp. right) semicentral idempotent;complete set of (centrally) primitive idempotents;
Language
English
Cited by
1.
Structure of Abelian rings, Frontiers of Mathematics in China, 2016
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