Trapdoor committement using ring lattices involving three parties
I think that this is just a digital signature scheme. $A$ should generate a private signing key and a public verification key for your signature scheme of choice. The then create a signature for the message $m$ and which you are calling the commitment $c(m)$. $B$ can then pass on the $c(m)$ to $C$ who can confirm that is is signed by $A$, but $B$ cannot produce signatures for other messages that verify using $A$'s verification key.
There are several lattice-based signature schemes though they are all quite tricky to parameterise and implement securely. The NIST competition has advanced the lattice-based signatures FALCON and Crystals DILITHIUM to the final round.