# 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 and are trivial for .

## 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*, 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 (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

- ↑ Template:Harvnb
- ↑
^{2.0}^{2.1}Template:Harvnb

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