TITLE: Ideal as a projective module
QUESTION [12 upvotes]: I am sorry, this may not be a good question here. I am looking a good reference about when the ideal $I$ of a given commutative ring $R$ (local or may not be local) with identity is a projective module.

REPLY [3 votes]: For rings in general (with identity), the ring is called right hereditary if all right ideals are projective, and right semihereditary if all finitely generated right ideals are projective. There are a lot of other interesting equivalences and properties that these rings have, which you can find here.
A commutative hereditary domain is the same thing as a Dedekind domain, and a commutative semihereditary domain is the same thing as a Prüfer domain.
For examples that aren't domains (usually), semisimple rings are hereditary (on both sides) and von Neumann regular rings are semihereditary (on both sides).