Hereditary ring

In mathematics, especially in the area of abstract algebra known as module theory, a ring "R" is called hereditary if all submodules of projective modules over "R" are again projective. If this is required only for finitely generated submodules, it is called semihereditary.
For a noncommutative ring "R", the terms left hereditary and left semihereditary and their right hand versions are used to distinguish the property on a single side of the ring. To be left (semi-)hereditary, all (finitely generated) submodules of projective "left" "R"-modules must be projective, and to be right (semi-)hereditary all (finitely generated) submodules of projective "right" "R"-modules must be projective. It is possible for a ring to be left (semi-)hereditary but not right (semi-)hereditary, and vice versa.