Primitive of holomorphic Function $\frac{1}{z}$ on an Annulus.
Solution 1:
Let $C$ be the circle of radius $1$ and center $(0,0)$, one has: $$\int_C\frac{\mathrm{d}z}{z}=\int_0^1\frac{2i\pi e^{2i\pi t}}{e^{2i\pi t}}\,\mathrm{d}t=2i\pi\neq 0.$$ If $z\mapsto 1/z$ has a primitive in $A(0,1,2)$, the above integral must be $0$. Whence the result.
Remark. I let you adapt the proof for $A(0,r,R)$ when $r<R$.