Is there a term for an "inverse-closed" subring of a ring?

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 $f$.)

Solution 1:

A ring extension $\rm\: R \subset S\:$ is said to be $\rm\:\cal C$-survival if every ideal $\rm\:I\:$ of type $\rm\:\cal C\:$ "survives" in $\rm\:S\:,\:$ i.e. $\rm\:I\ne R\ \Rightarrow\ I\:S \ne S\:.\:$ Your notion is the special case where $\cal\: C\:$ is the class of principal ideals, i.e. principal-survival. Such extensions are encountered in literature characterizing integral extensions in terms of various properties such as LO (lying-over), GO (going-up), INC (incomparbility), 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 Cokendall; Dobbs. Survival-pairs of commutative rings have the lying-over property.