= Solution
Solving the <Time-independent Schrödinger equation> with $\psi\propto e^{ik\phi}$ gives $E=\hbar^2k^2/(2mR^2)$. The periodic <boundary condition> requires $e^{2\pi ik}=1$, hence $k=n\in\mathbb Z$. With the angular integration measure $d\phi$, <wavefunction normalization> gives
$$
\int_0^{2\pi}|A|^2\,d\phi=1,
$$
so a convenient <orthonormal eigenbasis> is
$$
\boxed{u_n(\phi)=\frac{e^{in\phi}}{\sqrt{2\pi}},
\qquad E_n=\frac{\hbar^2n^2}{2mR^2},\qquad n\in\mathbb Z.}
$$
The $n=0$ <ground state> is nondegenerate. For each $|n|>0$, $u_n$ and $u_{-n}$ span a two-dimensional <energy eigenspace>; real <sine> and <cosine> combinations are an equally valid <basis>. These opposite-circulation <eigenstates> have <angular momentum> $n\hbar$ and $-n\hbar$.
If instead the <wavefunction> is normalized with respect to physical arc length $ds=R\,d\phi$, its expression is $e^{in\phi}/\sqrt{2\pi R}$. The two conventions describe the same <probability>, since $|u_n|^2d\phi=|u_n/\sqrt R|^2ds$. The remaining parts use $d\phi$.
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