Are finitely generated modules over a commutative local ring cancellative?

Question

Let $M,N,P$ be finitely generated modules over a (Notherian) local ring $R$. If $M \oplus N \cong M \oplus P$, do we have $N \cong P$ ? If not, what if we further assume that $M \cong R^n$ for some positive integer $n$, or even $M=R$ ?

Answer 1

This is true. See Proposition 1 in this paper which contains many other interesting results on decompositions of modules over local rings.

Update: Also see this paper of T.Y. Lam for a more recent reference (thanks to rschwieb for suggesting this).