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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 $f$.)

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Not an answer, but "inverse-closed subalgebra" seems well established in functional analysis. – Jonas Meyer Aug 15 '11 at 23:37
I am interested in this question because I also want to know: for a subset $X$ of a ring $R$, and a ring homomorphism $f$ from $R$ to another ring $T$, when $f(x)$ is determined for any $x \in X$, to what extend is $f$ determined? @Jonas Meyer: do you mean the inverse-closed subalgebra of a Banach algebra? The definition for this is similar to what Qiaochu Yuan wanted to define in a ring... I think the notion could be extended to rings (if it is not already done), because in my mind, algebras are special rings, and Banach algebras are special algebras. – ShinyaSakai Nov 13 '11 at 16:27
@ShinyaSakai: I agree, the case for algebras is a special case of the general case for rings. (The algebras I had in mind are usually, but not always, Banach algebras.) But even so that doesn't answer the question of what is or is not established terminology used by ring theorists. – Jonas Meyer Nov 14 '11 at 0:47
@Jonas Meyer: Yes. I think if the asker is writing a thesis, he might borrow the term from that of algebras :) – ShinyaSakai Nov 14 '11 at 11:19

A ring extension $\rm\: R \subset S\:$ is said to be $\rm\:\cal C$-survival if every ideal of $\rm\:I\:$ 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.

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