hereditary ring

Let R be a ring. A right (left) R-module M is called right (left) hereditary if every submoduleMathworldPlanetmath of M is projective over R.


  • If M is semisimplePlanetmathPlanetmathPlanetmath, then M is hereditary.

  • Suppose M is an external direct sum of hereditary right (left) R-modules, then M is itself hereditary.

A ring R is said to be a right (left) hereditary ring if all of its right (left) ideals are projective as modules over R. If R is both left and right hereditary, then R is simply called a hereditary ring.


  • Even though the notions of left and right heredity in rings are symmetrical, one does not imply the other.

  • If R is semisimple, then R is hereditary.

  • If R is hereditary, then every free R-module is a hereditary module.

  • A hereditary integral domainMathworldPlanetmath is a Dedekind domainMathworldPlanetmath, and conversely.

  • The global dimension of a non-semisimple hereditary ring is 1.

Title hereditary ring
Canonical name HereditaryRing
Date of creation 2013-03-22 14:48:50
Last modified on 2013-03-22 14:48:50
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 16D80
Classification msc 16E60
Defines hereditary module