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 submodules must be projective. It is possible for a ring to be left (semi-)hereditary but not right (semi-)hereditary, and vice versa.

Equivalent definitions

The ring R is left hereditary if and only if all left modules have projective resolutions of length at most 1. This is equivalent to saying that the left global dimension is at most 1. Hence the usual derived functors such as and are trivial for .