# A question about projective modules

These is an equivalent relation about projective modules. P is projective , (1)P is a direct summand of free module (2)If P is a quotient of the R-module M, then P is isomorphic to direct summand of M.

I am confused here, what does it mean that P is a quotient of the R-module M, the quotient module of M? Am I right with here?

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This means that if $\,P\cong M/N\,$ , then $\,P\,$ is isomorphic to a direct summand of o $\,M\,$ , or what is the same: if we have a short exact sequence
$$0\longrightarrow N\longrightarrow M\stackrel{\pi}\longrightarrow P\longrightarrow 0$$
then the sequence splits...which follows directly from the definition of projective, since then we can find a homom. $\,f:P\to M\,$ s.t. $\,\pi\circ f=Id_P\,$ .
Here the sequence spilits, that means $M=P\otimes N$, so M is a direct summand, but I can not see clearly how P is a direct summand – user53800 Jan 5 '13 at 0:38
No @user53800 , in $\,M=P\times N\,$ , it is $\,P\,$ which is a direct summand... – DonAntonio Jan 5 '13 at 0:40