I have to prove that if P is a R-module

redupticslaz 2022-04-27 Answered

I have to prove that if P is a R-module , P is projective right there is a family {xi} in P and morphisms fi:PR such that for all xP
x=iIfi(x)xi
where for each xP, fi(x)=0 for almost all iI.

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Answers (1)

August Moore
Answered 2022-04-28 Author has 17 answers
Step 1
Put the fi together to form one giant f from P to R(I), the direct sum of I copies of the ring R. The condition that
xfi(x)xi
just means that there is some g:R(I)P such that g(f(x))=x, namely g(r1 r2)=r1x1+r2x2+. In other words, P is a direct summand of the free module R(I)
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