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Let $R$ be a commutative ring (with unit). Then if $$0\longrightarrow M'\longrightarrow M\longrightarrow M''\longrightarrow 0$$ is an exact sequence of $R$-modules, with $M''$ $R$-flat, $M$ is flat if and only if $M'$ is flat.

Now suppose that $M'=M''$. The preceding result gives that $M$ is flat if $M'=M''$ is. My question is: if $M$ is flat, does this implies that $M'$ is flat ?

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If $M$ is flat, or even projective, and $M'$ is isomorphic to $M''$, it just implies that $M'$ is periodic. For example, if $R=k[t]/(t^2)$ and $M=R$ then we can have $M'=M''=k$.

You may also be interested in the fact that if $M'\cong M''$ is flat and $M$ is projective then $M'$ is projective. See a paper of Benson and Goodearl.

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