Flat module

A left $𝑅$ -module $𝑉$ is said to be flat iff any of the following equivalent conditions hold:

Proof

Suppose ^F1 holds. For any monomorphism $𝑓 \in 𝖬 𝗈 𝖽 𝑅 (𝐴, 𝐵)$ we have a short exact sequence

0 \to 𝐴 𝑓 ↪ 𝐵 ↠ 𝐵 / 𝐴 \to 0

whence

0 \to 𝐴 \otimes 𝑅 𝑉 𝑓 \otimes 𝑅 𝑉 ↪ 𝐵 \otimes 𝑅 𝑉 / 𝐴 ↠ 𝐵 / 𝐴 \otimes 𝑅 𝑉 \to 0

so $𝑓 \otimes 𝑅 𝑉$ is also a monomorphism, so ^F2 holds.

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