All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate multiplication von Neumann regular modules. For this reason we introduce the concept of nilpotent submodules generalizing nilpotent ideals and then prove that a faithful multiplication module is von Neumann regular if and only if it has no nonzero nilpotent elements and its Krull dimension is zero. We also give a new characterization for the radical of a submodule of a multiplication module and show in particular that the radical of any submodule of a Noetherian multiplication module is a finite intersection of prime submodules.

I am Professor Majid M. Ali, born in Iraq on 31 March, 1959. I am New Zealand citizen since 2000. I got my PhD degree from Universitat Kaiserslautern in March, 1994. I have worked in Auckland University, 1998-2001. Then I moved to Oman where I am working at the department of Mathematics and Statistics, Sultan Qaboos University.