A decomposition theorem for semiprime rings

Abstract

A ring \(A\) is called an \(FDI\)-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 \(FDI\)-ring is a direct product of a semisimple Artinian ring and a semiprime \(FDI\)-ring whose identity decomposition doesn't contain artinian idempotents.