Is there a classification of the commutative rings (with unit) such that each module over the ring is projective ?
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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. |
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