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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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1 Answer 1

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. $\,\pi\circ f=Id_P\,$ .

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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
    
oh I see, P is a direct summand as a component, is that right? I think I think it in a wrong way –  user53800 Jan 5 '13 at 0:42
    
Indeed @user53800, though perhaps it is more comment the name of "factor" or "direct, free, whatever factor" –  DonAntonio Jan 5 '13 at 0:43
    
Thanks so much, this has confused me for a long time, because I always get wrong result by using that, :) –  user53800 Jan 5 '13 at 0:46

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