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.
Two $R$-modules $M$ and $N$ are said to be "stably isomorphic" (see Stacks for example) if $M \oplus R^n$ and $N \oplus R^n$ are isomorphic for some $n \in \mathbb{N}$. It is possible for two non-isomorphic modules to be stably isomorphic. If $M'$ and $N'$ are such modules, let $n$ be the smallest (necessarily positive) integer such that $M' \oplus R^n \cong N' \oplus R^n$; then $M = M' \oplus R^{n-1}$ and $N = N' \oplus R^{n-1}$ satisfy your condition.
In The K-book: an introduction to algebraic K-theory, Weibel gives an example (Chapter I, Example 1.2.2) of a stably free module which is not free. The ring is $R = \mathbb{R}[x,y,z]/(x^2+y^2+z^2=1)$ and the module is $P = \ker \sigma$, where $\sigma : R^3 \to R$ is $(P,Q,R) \mapsto xP+yQ+zR$. Then $P \oplus R \cong R^3 = R^2 \oplus R$ as $R$-modules, but $P$ is not isomorphic to $R^2$. (Crazily enough, the reason comes from algebraic topology: it's the hairy ball theorem!)
In fact if you go through the definition, you will see that two finitely generated projective modules $M$, $N$ over $R$ are stably isomorphic iff they are equal in $K_0(R)$, the zeroth algebraic K-theory group of $R$. (See also the K-book, Chapter 2.)