Clean Modules Constructed from the External Direct Sum of Clean Modules

Let  be a ring with identity and  be an -module. A ring in which every element can be expressed as the sum of an idempotent and a unit element is called a clean ring. An -module  that is mapped to itself is called an endomorphism, denoted by . The set of all endomorphisms forms a ring under addition and function composition. This fact motivates the notion of clean modules over a ring. An -module  is called a clean module if  is a clean ring. This concept was first investigated by Camillo et al. In this article, we identify a special case of a clean module constructed from the external direct sum of two modules over the same ring. We show that if   and  are clean modules over , then their external direct sum  is also a clean module.