semihereditary ring

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

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


  • A hereditary ring is clearly semihereditary.

  • A ring that is left (right) semiheridtary is not necessarily right (left) semihereditary.

  • If R is hereditary, then every finitely generated submodule of a free R-modules is a projective moduleMathworldPlanetmath.

  • A semihereditary integral domain is a Prüfer domain, and conversely.

Title semihereditary ring
Canonical name SemihereditaryRing
Date of creation 2013-03-22 14:48:55
Last modified on 2013-03-22 14:48:55
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 16D80
Classification msc 16E60
Defines semihereditary module