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 Rmodule is projective. 2) Any right Rmodule is injective. 3) Any simple right Rmodule is projective. 4.1) Any right Rmodule 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. 

