1

I have given two modules $M$ and $N$ over a local ring $R$. I also know that $M \oplus N \cong R^n$ for some $n\in \mathbb{N}$. I then have to prove that both $M$ and $N$ are free modules.

Since $M \oplus N \cong R^n$ I know that $M \oplus N$ is free. If $M$ and $N$ are free it's easy to prove that $M \oplus N$ is free as well, however the opposite doesn't always hold. Therefore I have to do it some other way. I have tried constructing bases for the modules, but this was unsuccessful. Can anyone please help me?

MarlonButBetter
  • 139
  • 3
  • I have collected some links where the same question has been asked before. They are easy to find with the search function - once you know the correct search term is "projective module over local ring". – Martin Brandenburg Nov 22 '23 at 18:11

0 Answers0