  Topologically ordered states are fundamentally important in theoretical
physics, which are also suggested as promising candidates to build
fault-tolerant quantum devices. However, it is still elusive how topological
orders can be affected or detected under noises. In this work, we find a
quantity, termed as the ring degeneracy $\mathcal{D}$, which is robust under
pure noise to detect both trivial and intrinsic topological orders. The ring
degeneracy is defined as the degeneracy of the solutions of the self-consistent
equations that encode the contraction of the corresponding tensor network(TN).
For the $Z_N$ orders, we find that the ring degeneracy satisfies a simple
relation $\mathcal{D} = (N + 1)/2 + d$, with $d = 0$ for odd $N$ and $d = 1/2$
for even $N$. Simulations on several non-trivial states (two-dimensional Ising
model, $Z_N$ topological states, and resonating valence bond states) show that
the ring degeneracy can tolerate noises up to a strength associated to the gap
of the TN boundary theory.
