Generalized Morphic Group Rings

Let A be a commutative ring with identity, G be an abelian group, and consider the group ring AG. A ring A is called a generalized morphic ring (GM ring) if the annihilator of each element in A is principal. In this article, we showed that if AG is a GM ring, then so is A. The converse was proved to be false. We try to put some conditions on A or G to get the converse. Among many other results, we showed that if A is an Armendariz ring and G is a torsion free group, then AG is a GM ring if and only if A is.