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 similarly to be right (semi-)hereditary all (finitely generated) submodules of projective right R-modules must be projective.
This abstract algebra-related article is a stub.