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.

• 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 domain is a Dedekind domain, and conversely.

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