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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.

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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 $S$.

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  • $\begingroup$ See also Paul Cohn's inert extensions in the paper linked here. Iirc later authors studied variations on that. $\endgroup$ – Bill Dubuque Jul 13 '19 at 3:52
  • $\begingroup$ @BillDubuque yea it's definitely reminiscent of inert extensions - 'half inert,' I guess. One thing worth mentioning (which I didn't notice two years ago) is that any faithfully flat morphism of a domain has this property. $\endgroup$ – Badam Baplan Oct 6 '19 at 23:34
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Never heard of it.

And if you allow $x=0$, then the rationals don't have this relationship with the reals, nor does any proper ring extension.

If you restrict to $x\neq 0$, then obviously any field extension has this property.

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  • $\begingroup$ Yea good point, giving a field as an example was pointless. The property becomes meaningful when looking at rings that aren't division rings, and more so looking at non UFDs (although the example i gave is still totally trivial). $\endgroup$ – Badam Baplan Nov 17 '17 at 23:24

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