Hereditary Group Rings

The purpose of this note is to describe those group rings that are right hereditary. The characterization necessarily involves a number of concepts from ring theory and from group theory, and we briefly review these for the benefit of the reader. From ring theory we need the following definitions. A ring R is said to be: right hereditary if every right ideal is projective (as right R ‐module); completely reducible if R is a finite direct product of full matrix rings over skew fields (or equivalently, R is nonzero and every right ideal of R is a direct summand of R ); von Neumann regular if every right R ‐module is flat (or equivalently, for each element r of R there exists an element x of R such that rxr = r ); right ℵ 0 ‐ Noetherian if every right ideal of R is countably generated.