TITLE: $S$ a subring of $R$ has the property that if $x,y \in S$, $y \not= 0$ and $xz = y$ in $R$ then $z \in S$
QUESTION [5 upvotes]: I'm having trouble remembering if the property described in the title has an actual name.   
Phrased informally, if an element of $S \subset R$ partially factors in $S$, then that factorization is actually valid in $S$.
For example, the integers embedded in the rationals don't possess this property, but the integers embedded in $\mathbb{Z}[\sqrt 5])$ (or many other algebraic extensions of the integers) do, even though many integers have factorizations lying entirely in  $\mathbb{Z}[\sqrt 5 ]) \setminus \mathbb{Z}$.
To further illustrate, a slightly less trivial and rather artificial example of a ring with this property is the following: 
Let $R$ be a GCD domain, and consider the ring of polynomials over $R$, let's call it $R_{gcd}[X]$, in which multiplication is normal polynomial multiplication and addition is defined on the monomial basis as $aX^i + bX^i \rightarrow \gcd(a,b)X^i$.  Fix an $r \in R$ not a unit.  Gauss' lemma goes to show that the polynomials whose content has a factorization as a pure power of $r$ form a subring of $R_{gcd}[X]$, and moreover this subring  is easily seen to have the property described above.  Furthermore, if $R$ is not a UFD (say, the algebraic integers), then for some choices of $r$ there will exist factorizations of polynomials in the subring entirely in terms of polynomials not in the subring.

REPLY [2 votes]: I found this paper which calls a subring $S$ of $R$ that has the above property self-factorially closed and provides some discussion of equivalent conditions and implications.  The stronger notion of $xy \in S \implies x,y \in S$ is there called factorially closed.  Note that the analogous nomenclature for multiplicative subsets of a ring is more universally known as saturation.  
If $S$ is self-factorially closed in $R$, with $T(S)$ the field of fractions of $S$, then we have the identity $$S = T(S) \cap R$$ we can in some cases be useful to see that properties of $R$ (e.g. integral closure) are inherited by $S$.