13

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 $f$.)

Bill Dubuque
  • 272,048
Qiaochu Yuan
  • 419,620
  • 2
    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 00:47
  • @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

2 Answers2

10

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. $2003$.

Bill Dubuque
  • 272,048
  • You seem to be using that $\langle r \rangle = \langle 1 \rangle$ iff $r$ is a unit. But this may fail for non-commutative rings. In fact, if $r$ is left-invertible, we already have $\langle r \rangle = \langle 1 \rangle$. In general, $\langle r \rangle$ is additively generated by products $u \cdot r \cdot v$ for $u,v \in R$. Also, even in the commutative case (where the equivalence holds), the term "principal-survival" is not a satisfactory answer since it does not explicitly speak about units: a reader may not think about units in the first place. "inverse-closed" is better. – Martin Brandenburg Nov 26 '23 at 20:11
  • @Martin The question is tagged commutative-algebra. Said terminology is commonly used by ring theorists studying these matters - see the cited paper and its citations. – Bill Dubuque Nov 26 '23 at 20:19
  • Ah ok I didn't see that. I just read "ring $R$" in the question and was assuming that the rings are not necessarily commutative. – Martin Brandenburg Nov 26 '23 at 20:20
  • 1
    erm ... I just saw that originally the question only had the "ring theory" tag (besides "terminology"). In 2022, for some reason, you added the "commutative algebra" tag. This is a bit fishy. – Martin Brandenburg Nov 26 '23 at 21:32
  • @Martin I didn't recall that, but the OP didn't object to the tag, so perhaps they were also considering the much more common commutative case. In any case the tag is helpful for searches. – Bill Dubuque Nov 26 '23 at 21:57
  • I really don't want to argue, but I don't agree. You added a tag which was probably not intended when the question was asked (after all, it did not ask for commutative rings in the body), and your question only refers to the commutative case, without even mentioning this restriction (even not after my comment). Also, as my answer and Jonas' comments shows, the non-commutative is also very common. I suggest that you edit your question so that every reader (this question pops up easily on Google) will know that you are talking about commutative rings. – Martin Brandenburg Nov 26 '23 at 22:07
  • @Martin One could make the same argument about thousands of questions - where what the OP intended could go either way. It's not a fruitful discussion (and does not belong under some answer). In any case, I have added the noncommutative-algebra tag too so searches on either will work here (not an optimal solution, but we are constrained by the primitive SE platform). – Bill Dubuque Nov 26 '23 at 22:27
1

Bill Dubuque gave an answer in the commutative case. For the general case, we can just say, well, "closed under taking inverses", but in functional analysis the term inverse closed (also inverse-closed) is common to denote Banach subalgebras with that property. See for example:

Bruce A. Barnes, Inverse Closed Subalgebras and Fredholm Theory, Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, Vol. 83A, No. 2 (1983), pp. 217-224 (8 pages)

Karlheinz Gröchenig, Michael Leinert, Inverse-Closed Subalgebras of Noncommutative Tori, arXiv:1208.6229

Since the property just refers to the underlying rings (actually, just the underlying multiplicative monoids) of the Banach algebras in consideration, it does make sense to use the same terminology for general rings. And indeed, the terminology inverse-closed subrings is used in a few papers, such as:

Dragana Cvetkovic-Ilic, Robin Harte, The spectral topology in rings, Studia Mathematica 200 (3) (2010)

Beren Sanders, Higher comparison maps for the spectrum of a tensor triangulated category, Advances in Mathematics 247, pp. 71-102

The property is also studied in the paper

Jeno Szigeti, Leon van Wyk, Subrings which are closed with respect to taking the inverse, Journal of Algebra 318(2)

It is remarkable that they do not use any terminology (well, except for the one in the title). They just repeat the condition $S \cap U(R) = U(S)$ over and over again (sometimes with elements).

Martin Brandenburg
  • 163,620
  • 1
    +1 Good to have some literature references for the noncommutative case. Were these found by searching databases and - if so - was there only a few matches, or is the use widespread? – Bill Dubuque Nov 26 '23 at 22:30
  • I used google with very basic search terms, I don't have access to any databases (anymore). Jonas already mentioned that the terminology is common for Banach algebras, which I can confirm. It is much less common for rings (not saying that there is any other terminology: I just didn't find so many papers), but I don't know why. – Martin Brandenburg Nov 26 '23 at 22:50