Zero Insertive Group Rings

The aim of this research is to find the necessary and sufficient conditions on a ring A and a group G for which the group ring A[G] to be a zero inserted ring (zi-ring), a zero commutative ring (zc-ring), and a duo- ring. In this paper, we found that the necessary conditions for A[G] to be zi-ring are that G must be Hamiltonian group and A is zi-ring whenever G is a periodic group. Also similar results for A[G] to be zc-ring and duo-ring are given.