Semihereditary ring

Remarks

Example: every hereditary ring is semihereditary, since the condition is asked only of some of the left ideals. A non-hereditary example is the ring $\overline{\Z}$ of all algebraic integers. It is a Bézout domain, so every finitely generated ideal is principal, hence free of rank $0$ or $1$, hence projective. It is not Noetherian, because $2^{1/2}, 2^{1/4}, 2^{1/8}, \dots$ generate a strictly increasing chain of principal ideals; and a hereditary domain is a Dedekind domain, which is Noetherian. Non-example: the ring $\Z/4\Z$ is not semihereditary, because its ideal $2\Z/4\Z$ is finitely generated and not projective. Equivalent form: $R$ is left semihereditary if and only if every finitely generated submodule of a projective left $R$-module is projective. A commutative integral domain is semihereditary if and only if it is a Prüfer domain.