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 (semi-)hereditary, all (finitely generated) submodules of projective leftR-modules must be projective, and similarly to be right (semi-)hereditary all (finitely generated) submodules of projective rightR-modules must be projective. It is possible for a ring to be left (semi-)hereditary but not right (semi-)hereditary and vice versa.
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.
An important example of a (left) hereditary ring is the
path algebra of a
quiver. This is a consequence of the existence of the standard resolution (which is of length 1) for modules over a path algebra.
The
triangular matrix ring is right hereditary and left semi-hereditary but not left hereditary.
If S is a von Neumann regular ring with an ideal I that is not a direct summand, then the triangular matrix ring is left semi-hereditary but not right semi-hereditary.
Properties
For a left hereditary ring R, every submodule of a
free left R-module is
isomorphic to a direct sum of left ideals of R and hence is projective.[2]