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.

• 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 module.

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