In mathematics, especially in the area of abstract algebra known as module theory, a ringR 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 ringR, 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 leftR-modules must be projective, and to be right hereditary all submodules of projective right submodules must be projective. It is possible for a ring to be left hereditary but not right hereditary, and vice versa.
Equivalent definitions
The ringR is left hereditary if and only if all left ideals of R are projective modules.
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.
Examples
Semisimple rings are left and right hereditary via the equivalent definitions: all left and right ideals are summands of R, and hence are projective. By a similar token, in a von Neumann regular ring every finitely generated left and right ideal is a direct summand of R, and so von Neumann regular rings are left and right semihereditary.
For any nonzero element x in a domainR, via the map. Hence in any domain, a principal right ideal is free, hence projective. This reflects the fact that domains are right Rickart rings. It follows that if R is a right Bézout domain, so that finitely generated right ideals are principal, then R has all finitely generated right ideals projective, and hence R is right semihereditary. Finally if R is assumed to be a principal right ideal domain, then all right ideals are projective, and R is right hereditary.
A commutative hereditary integral domain is called a Dedekind domain. A commutative semi-hereditary integral domain is called a Prüfer domain.
An important example of a hereditary ring is the path algebra of a quiver. This is a consequence of the existence of the standard resolution for modules over a path algebra.
The triangular matrix ring is right hereditary and left semi-hereditary but not left hereditary.
