Give an example of a UFD having a subring which is not a UFD. Give an example of a UFD having a subring which is not a UFD.
I thought of $\mathbb{Z}[\sqrt{2},\sqrt{3}]$.
Could you please explain my question. I am trying grasp the concepts, need help.
 A: One question to ask after reviewing the two answers already posted is whether the existence of an extension $S\subset R$, with $R$ a UFD and the nonunits of $S$ being nonunits in $R$, makes $S$ a UFD.
While plausible, this is also false. For instance $k[x^2,x^3]\subset k[x]$ is a counterexample for any UFD $k$. This is because $x^2$ and $x^3$ are irreducible and yet $x^6=(x^2)^3=(x^3)^2$.
A: $\mathbb{Z}[2i]$ as a subset of $\mathbb{Z}[i]$. First one is not integrally closed, so can't be UFD, second is Euclidean.
A: Take any integral domain which is not a UFD and consider it as a subring of its field of fractions. (Fields are UFD for trivial reasons and if you don't accept this, take the polynomial ring over it)
A: Maybe I am beating the dead horse, but...
$\mathbb{Q} + X \mathbb{R}[X] \subset \mathbb{R}[X]$
is one of many similar examples.
A: Consider $\mathbb C$ , it is a field hence obviously $UFD $ but if you consider a subring $\mathbb Z[\sqrt {-5}]$, it is not a UFD . In fact,
$9=3.3$ and also $9= (2+\sqrt{-5} ) . (2-\sqrt{-5})$ , Hence the factorization is not unique.
