English: Properties of modules in commutative algebras and implications between them.

References:

free => projective: projective modules are exactly direct summands of free ones: Lang III.4, p. 137.
projective => flat: same reason, tensor products commute with direct sums.
flat => torsion-free: torsion is the kernel of M tensor (A -> Q) if Q is the total quotient ring, M the module, A the ring.
module torsion-free + ring Dedekind => module flat: Liu, Corollary 1.2.14
module flat and ring perfect => module projective: wikipedia page for perfect rings
module projective + ring local => module free: Matsumura, Theorem 2.5

module projective + ring PID => module free: Lang, Theorem III.7.1, p. 146 and App. 2.2, p. 880