Exercise 9, § 1

Let P be an a-module. Prove that the following conditions are equivalent: α) For a sequence M' $\twoheadrightarrow$ M $\twoheadrightarrow$ M'' of right a-modules to be exact, it is necessary and sufficient that the sequence

be exact. β) P is flat, and for every nonzero right a-module M, one has $M \otimes_A P \neq (0)$. γ) P is flat, and for all right a-modules M, N, the canonical homomorphism is injective. δ) P is flat, and for every maximal right ideal m of A, one has $mP \neq P$. P is then said to be a faithfully flat A-module.