# 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 (semi-)hereditary if and only if all (finitely generated) left ideals of R are projective modules.[1][2]
• The ring R is left hereditary if and only if all left modules have projective resolutions of length at most 1. Hence the usual derived functors such as $\mathrm{Ext}_R^i$ and $\mathrm{Tor}_i^R$ are trivial for $i>1$.

## Examples

• Semisimple rings are easily seen to be 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 domain R, $R\cong xR$ via the map $r\mapsto xr$. 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.

## 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]

## References

1. ^ Lam 1999, p. 42
2. ^ a b Reiner 2003, pp. 27–29