A decomposition theorem for semiprime rings

Abstract

A ring A is called an F DI-ring if there exists a decomposition of the identity of A in a sum of finite number of pairwise orthogonal primitive idempotents. We call a primitive idempotent e artinian if the ring eAe is Artinian. We prove that every semiprime F DI-ring is a direct product of a semisimple Artinian ring and a semiprime F DI-ring whose identity decomposition doesn’t contain artinian idempotents.