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 $f$.)