semihereditary ring
Let $R$ be a ring. A right (left) $R$module $M$ is called right (left) semihereditary if every finitely generated^{} submodule^{} 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.
Remarks.

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A hereditary ring is clearly semihereditary.

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

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If $R$ is hereditary, then every finitely generated submodule of a free $R$modules is a projective module^{}.

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A semihereditary integral domain is a Prüfer domain, and conversely.
Title  semihereditary ring 

Canonical name  SemihereditaryRing 
Date of creation  20130322 14:48:55 
Last modified on  20130322 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 