Finiteness, uniqueness and isomorphic properties in decompositions of group rings R[G]

We have found that any finite decompositions of group rings R[G] algebraic structure depend upon three properties. These properties are finiteness, uniqueness as well as isomorphism. Without utilizing these three properties we cannot decompose any algebraic structure finitely. We have first used many lemmas, propositions and theorems to find out the decompositions in group rings R[G]. Let us suppose that G be a group. As we decompose it as a finite number of two decompositions, as follow.