When a group ring is hereditary (Dicks's theorem)

Say $G$ is $R$ torsion-free if the orders of finite subgroups of $G$ are units in $R$. The group ring $R[G]$ is right hereditary if any of the following occur. 1) $R$ is semisimple and $G$ is the fundamental group of a connected graph of finite $R$-torsion free groups; 2) $R$ is right $\aleph_0$-Noetherian, von Neumann regular and $G$ is a countable, locally-finite $R$ torsion-free group; 3) $R$ is right hereditary and $G$ is a finite $R$ torsion-free group.


  • W. Dicks. Hereditary group rings. (1979) @ Theorem 1 p. 27