Classical Artinian Module and Related Topics

An R-module M is Artinian iff every non-empty set of submodules admits a minimal element. The aim of this paper, is studied three cases. The first case is to provide general properties about Artinian module. The second case is explaining the relationship between semisimple module and Artinian module and the third case is the study of Artinian module over division ring. In a short way, Artinian modules are characterized by the existence of minimal elements. This suggests a close analogy between Artinian module and other concepts. We proved that if M is projective, then it is Artinian. Also every Division module over Division ring is Artinian module. Any non-zero Sub-mod N of sem-simple R-module. If N is a non-zero Sub-mod of M, then N is Artinian as a module.