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2 edited body

In the interest of having an undeleted answer, here is a small result. Let $x, y$ be objects and $f, g : x \to y$ and $u, v : y \to x$ be morphisms in $C$, and let

$$F = af + bg, G = cu + dv$$

be two morphisms in $RC$, where $a, b, c, d \in R$. If $FG = \text{id}_x, text{id}_y, GF = \text{id}_y$, text{id}_x$, then WLOG$fu = \text{id}_x$text{id}_y$ and also some term in $GF$ must equal $\text{id}_y$. \text{id}_x$. If we want$x, y$to be non-isomorphic, then$f$cannot have a right left inverse and$u$cannot have a left right inverse, so it must be the case that$vg = \text{id}_y$text{id}_x$ and moreover no other composition of morphisms except $fu$ or $vg$ can be an identity.

It follows that $ac = bd = 1$, hence $a, b, c, d$ are all units, so none of the four terms in $FG$ or in $GF$ vanish. Thus the only way for all of the non-identity terms to cancel is if $gu = fv = gv$ and $ug = vf = vg$. But this implies

$$gug = fvg = f = gvg = g$$

and symmetrically $u = v$, so in fact $x, y$ must be isomorphic in $C$. Next on the list is linear combinations of three morphisms...

1

In the interest of having an undeleted answer, here is a small result. Let $x, y$ be objects and $f, g : x \to y$ and $u, v : y \to x$ be morphisms in $C$, and let

$$F = af + bg, G = cu + dv$$

be two morphisms in $RC$, where $a, b, c, d \in R$. If $FG = \text{id}_x, GF = \text{id}_y$, then WLOG $fu = \text{id}_x$ and also some term in $GF$ must equal $\text{id}_y$. If we want $x, y$ to be non-isomorphic, then $f$ cannot have a right inverse and $u$ cannot have a left inverse, so it must be the case that $vg = \text{id}_y$ and moreover no other composition of morphisms except $fu$ or $vg$ can be an identity.

It follows that $ac = bd = 1$, hence $a, b, c, d$ are all units, so none of the four terms in $FG$ or in $GF$ vanish. Thus the only way for all of the non-identity terms to cancel is if $gu = fv = gv$ and $ug = vf = vg$. But this implies

$$gug = fvg = f = gvg = g$$

and symmetrically $u = v$, so in fact $x, y$ must be isomorphic in $C$. Next on the list is linear combinations of three morphisms...