Hereditary ring

Remarks

Example: the ring $\Z$ is hereditary. Each ideal is $n\Z$ for some $n \ge 0$; it is $0$ or isomorphic to $\Z$ through $x \mapsto x/n$, so it is free and hence projective. Non-example: the ring $\Z/4\Z$ is not hereditary. It is a local ring, so its projective modules are free. The ideal $2\Z/4\Z$ has $2$ elements, while a nonzero free $\Z/4\Z$-module has at least $4$, so this ideal is not projective. Equivalent forms: $R$ is left hereditary if and only if every submodule of a projective left $R$-module is projective. This holds if and only if the left global dimension of $R$ is at most $1$. For a ring that is Noetherian on both sides, left and right hereditary are equivalent. An integral domain is hereditary if and only if it is a Dedekind domain. The path algebra of a finite quiver without oriented cycles over a field is hereditary.