Dunwoody's theorem on projectivity of the augmentation ideal
If the augmentation ideal of $R[G]$ is projective then $G$ is the fundamental group of some connected graph of finite groups having no $R$-torsion. (First proven for commutative rings, but indeed true in general.)
Reference(s)
M. J. Dunwoody. Accessibility and groups of cohomological dimension one. (1979) @ (location needed)
W. Dicks. Hereditary group rings. (1979) @ Theorem 8 p 35