Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question

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

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.