What is 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 hereditary, all submodules of projective left R-modules must be projective, and similarly to be right hereditary all submodules of projective right R-modules must be projective. It is possible for a ring to be left hereditary but not right hereditary and vice versa.