STRUCTURE OF IDEMPOTENTS IN POLYNOMIAL RINGS AND MATRIX RINGS

An idempotent e of a ring R is called right (resp., left) semi-central if er = ere (resp., re = ere) for any r is an element of R, and an idempotent e of R\{0, 1} will be called right (resp., left) quasicentral provided that for any r is an element of R, there exists an idempotent f = f(e, r) is an element of R\{0, 1} such that er = erf (resp., re = f re). We show the whole shapes of idempotents and right (left) semicentral idempotents of upper triangular matrix rings and polynomial rings. We next prove that every nontrivial idempotent of the n by n full matrix ring over a principal ideal domain is right and left quasicentral and, applying this result, we can find many right (left) quasicentral idempotents but not right (left) semicentral.