Systems Simplicity

Abstract

A simple system is a system who has no proper ideals. We prove that every simple  system  $\mathcal{J}$ have one of the following assertion:
 \begin{description}
   \item[$(1)$] $\mathcal{J}$  is $\mathfrak{h}-$irreducible.
   \item[$(2)$] $\mathcal{J}=\mathcal{J}_1\bigoplus\widetilde{\mathcal{J}_1}$ is the direct summation of two $\mathfrak{h}-$invariant   and $\mathfrak{h}-$irreducible subsystems.
   \end{description}