Suppose $M$ and $N$ are non-isomorphic $R$-modules (where $R$ is a commutative ring with a unit element).Can we conclude that $R \oplus M \not\simeq R \oplus N$ ? If not in this most general setup, is there an affirmative answer in some special circumstances ?
Motivation : I am reading Proposition 9(1.4) in Dale Husemoller's book 'Fibre Bundles' which reads :
If $u,v : \theta^1 \to \xi^k$ are two monomorphisms of the trivial line bundle over $B$ into $\xi^k$ (a k-dimensional vector bundle over $B$) , such that $n \leq ck-2$, then coker $u$ and coker $v$ are isomorphic over $B$.
Using the fact that short exact sequences of finite dimensional vector bundles over a paracompact space split, and replacing vector bundles over $B$ by $R$-modules, I get the question that I am asking here.
Essentially I am just trying to see if the theorem about vector bundles can be generalized to an arbitrary abelian category.