SEMICENTRAL IDEMPOTENTS IN A RING

Abstract

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