hereditary ring
Let $R$ be a ring. A right (left) $R$module $M$ is called right (left) hereditary if every submodule^{} of $M$ is projective over $R$.
Remarks.

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If $M$ is semisimple^{}, then $M$ is hereditary.

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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.
Remarks.

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

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If $R$ is semisimple, then $R$ is hereditary.

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If $R$ is hereditary, then every free $R$module is a hereditary module.

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A hereditary integral domain^{} is a Dedekind domain^{}, and conversely.

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The global dimension of a nonsemisimple hereditary ring is 1.
Title  hereditary ring 

Canonical name  HereditaryRing 
Date of creation  20130322 14:48:50 
Last modified on  20130322 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 