# Is this a valid proof for why all modules over a field are projective?

Let $F$ a field and $V$ an $F$-module (i.e. an $F$-vector space). Then $V$ has a basis, and so $V$ is free. All free modules are projective, so $V$ is projective too.

Is this ok?

• Yes, it is OK. The main point of course is this question. – Dietrich Burde Dec 17 '17 at 19:52
• "ok proof" = "proof"... – Jean Marie Dec 17 '17 at 19:53
• @JeanMarie I of course meant "ok" as in "valid" – Alex Dec 17 '17 at 19:54
• As a generalization, all modules over a division ring are also projective. – stressed out Dec 17 '17 at 20:33
• Having all modules projective characterizes semisjmple rings. Having all modules free characterizes division rings. – rschwieb Dec 17 '17 at 22:37