Hereditary ring

Summary

A left hereditary ring is a ring in which every left ideal is a projective module. Equivalently, every submodule of a projective left module is projective, so the left global dimension is at most $1$. Principal ideal domains, Dedekind domains and path algebras over a field of finite quivers without oriented cycles are hereditary. These rings are the next simplest case after semisimple rings, where every module is projective.