Is there a classification of the commutative rings (with unit) such that each module over the ring is projective ?
They're called "semisimple artinian" rings. Prove that a ring $R$ (no commutativity is required) is semisimple artinian iff (equivalently)
0) (definition is most books in Ring Theory) $R$ is right artinian and has no nonzero nilpotent right ideals.
1) Any right R-module is projective.
2) Any right R-module is injective.
3) Any simple right R-module is projective.
4.1) Any right R-module is semisimple
4.2) R is a semisimple right module over itself (if you want, $R_R$ equals its socle).
5) $R$ consists of the sum of (finitely many) right ideals.