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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.

If $M$ is semisimple, 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.
Remarks.

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

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

A hereditary integral domain is a Dedekind domain, and conversely.

The global dimension of a nonsemisimple hereditary ring is 1.
Mathematics Subject Classification
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