Projective implies flat

Every projective module is flat, so tensoring with it preserves exact sequences.

Projective implies flat

Projective implies flat: Let $P$ be a projective right $R$ -module. Then $P$ is flat: for every short exact sequence of left $R$ -modules

0\to A\to B\to C\to 0,

the induced sequence

0\to A\otimes_R P\to B\otimes_R P\to C\otimes_R P\to 0

is exact.

A standard proof uses that projective modules are direct summands of free modules , and that tensoring preserves exactness for free modules and respects direct summands, yielding flatness .