For a nonunital ring $R$ (or "rng") one has to be a little bit careful when considering the category of (left or right) $R$-module and, furthermore, the standard equivalent definitions of projective modules are in general not equivalent anymore (see e.g.https://math.stackexchange.com/questions/120458/projective-modules-over-rings-without-unit). To fix terminology: by projective module we mean the standard categorical definition https://en.wikipedia.org/wiki/Projective_module.

Let $M_n(R)$ denote the set of $(n\times n)$-matrices with entries from $R$ and let $e\in M_n(R)$ and $f\in M_m(R)$ be idempotents. It is easy to show that $M=eR^n$ and $N=fR^m$ are projective modules. For a *unital* ring, the statement that $M$ is isomorphic to $N$ is equivalent to saying that there exists $u\in GL_k(R)$ such that $e=ufu^{-1}$ (tacitly assuming an appropriate $k$ and an embedding for $e$ and $f$ into $M_k(R)$). Now, what is the corresponding statement for nonunital rings?

The usual algebraic definition of equivalence of idempotents state that $e\sim f$ if there exists $x,y$ such that $e=xy$ and $f=yx$. It is easy to see that this implies that $M\simeq N$ (multiplication by $y$ gives the isomorphism $M\to N$). Now, is the converse true? That is, if $M\simeq N$ then there exists $x,y$ such that $e=xy$ and $f=yx$?

The standard proof for unital rings uses the fact that a projective module is a direct summand of a free module, which may no longer be true for nonunital rings.