Journal of the
Korean Mathematical Society JKMS

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. $0 \neq e \in S_{r}(R)$).

Keywords: left (resp. right) semicentral idempotent, complete set of (centrally) primitive idempotents

MSC numbers: 17C27