SEMICENTRAL IDEMPOTENTS IN A RING

Let R be a ring with identity 1, I(R) be the set of all nonunit idempotents in R and $S_{\ell}$(R) (resp. $S_r$(R)) be the set of all left (resp. right) semicentral idempotents in R. In this paper, the following are investigated: (1) $e{\in}S_{\ell}(R)$ (resp. $e{\in}S_r(R)$) if and only if re=ere (resp. er=ere) for all nilpotent elements $r{\in}R$ if and only if $fe{\in}I(R)$ (resp. $ef{\in}I(R)$) for all $f{\in}I(R)$ if and only if fe=efe (resp. ef=efe) for all $f{\in}I(R)$ if and only if fe=efe (resp. ef=efe) for all $f{\in}I(R)$ which are isomorphic to e if and only if $(fe)^n=(efe)^n$ (resp. $(ef)^n=(efe)^n$) for all $f{\in}I(R)$ 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 $0{\neq}e{\in}S_{\ell}(R)$ (resp. 0$0{\neq}e{\in}S_r(R)$).