Is it true that when a UFD is another ring $R$'s ideal, then ring $R$ is also a UFD?
I find an example but I'm not sure: the holomorphic ring $\mathcal O_x$, it's a UFD and the meromorphic ring $\mathcal M_x$, it's also a UFD (I guess but I'm not sure).
If your ideal shared identity with the containing ring, then trivially yes, since an ideal containing the identity is the entire ring.
If it is an ideal with a different identity, then no because the identity is a nontrivial idempotent, and domains don’t have nontrivial idempotents.