Group rings and the importance of the subject

In this study, we deal with the concept of group ring. We survey some case which group ring has no zero divisors. We define a map tr, from the group ring K[G] into the field K. Let e? K[G] be an idempotent and we have shown that tre is contained in the prime supfield of K. We mention some condition which the group ring K[G] is prime. Let C be field of complex numbers, we say that for all group rings G,C[G] is semisimple. Let G be finite group, we obtain homorphisms of the group ring R[G]. We define derivation in the group ring and in the free group ring and we establish a theorem which we use applications of Fox?s derivation and we have some examples with applications of this theorem. Let G be a finitely generated abelian group, we compute the group of units of the integral group ring Z(G) and the group of automorphisms of Z(G). Furthermore, we mention hypothesis in the group ring