Recovering Unique Factorization

Question

Can a (commutative) ring $S$ fail to have unique factorization but be a subring of a (possibly noncommutative) ring $R$ which does have unique factorization?

The idea being that the irreducibles in $S$ split further when viewed as a subring of $R$, thus somehow recovering unique factorization. It probably makes sense to require that the notion of unique factorization in $R$ reduces to the one in $S$.

Answer 1

Take any domain which is not a UFD; then it is a subring of its field of fractions, which is a UFD.

Answer 2

$S = \mathbb Z[\sqrt{-5}] \subseteq \mathbb C = R$ or $S = K[x^2,x^3] \subseteq K[x] = R$.

Answer 3

Yes, a non-integrally closed subring of a UFD (e.g. $\,\Bbb Z[\sqrt{-3}]\subset \,\Bbb Z[(-1\!+\!\sqrt{-3})/2)\,$ is not a UFD, since UFDs are integrally closed (by the monic case of Rational Root Test).

Remark $\ $ Your idea of enlarging the ring to recover unique factorization is at the heart of Kummer's method of introducing "ideal numbers" in order to refine nonunique factorizations into unique factorizations. This was generalized in Kronecker's divisor theory (and, in another direction, by Dedekind's ideal theory).