I would like to know whether there are established terms for
- A subring $S$ of a ring $R$ such that $S \cap U(R) = U(S)$; in other words, every element of $S$ which is invertible in $R$ is invertible in $S$.
- The smallest subring $S$ of a ring $R$ containing some set $r_1, r_2, ...$ of elements of $R$ satisfying the above property.
Motivation: if $f : R \to T$ is a ring homomorphism, then knowing $f(r_1), f(r_2), ...$ implies that you know $f$ on the subring $S$ above. (Contrast the corresponding motivation for subrings: if $f : T \to R$ is a ring homomorphism, then knowing that $r_1, r_2, ...$ are in the image of $f$ implies that the subring generated by $r_1, r_2, ...$ is in the image of $f$.)
Yes. A (commutative) ring extension $ \: R \subset S\:$ is said to be $ \:\cal C$-survival if every ideal $ \:\!I\:\!$ of type $ \:\!\cal C\:\!$ survives in $ \,S,\,$ i.e. $ \:\! I\:\!$ doesn't blowup to $(1)$ when extended to $ \:\! S,\,$ i.e. $ \:I\ne R\Rightarrow IS \ne S.\:$ Your notion is the special case where $\:\!\cal C\:\!$ is the class of principal ideals, i.e. principal-survival.
This notion plays a key role in results characterizing integral extensions in terms of various properties such as LO (lying-over), GO (going-up), INC (incomparability), etc. For example, a ring homomorphism is integral (resp., satisfies LO) if and only if it is universally a survival-pair homomorphism (resp., universally a survival homomorphism); see the paper below.
Coykendall; Dobbs. Survival-pairs of commutative rings have the lying-over property. $2003$.