Talk:Unit (ring theory)

Untitled

Even though both articles are a bit stub-like, care should be taken before merging unit ring and unit (ring theory) :

• on one hand, a unit ring and a unit of a ring are different objects and merit IMHO seperate entries
• notice in particular that a unit (ring theory) is any invertible element, and not the unit (= 1) of the (unital) ring (!!!) - while unit ring refers to the unit (neutral for multiplication).
• the "(ring theory)" in the second one arose historically as a disambiguation from other meanings of unit - it could have been chosen to be "(algebra)" quite as well - b.t.w. unit (algebra) redirects here, maybe a bit misleadingly - since "unit (algebra)" could mean both, the 1 of a unital algebra, and any unit of an algebra, seen as a ring (and, also, the algebraic notion of neutral element for multiplication).

Personally, I'm rather against this merging. — MFH:Talk 03:49, 10 March 2006 (UTC)

I disagree with merger also, these articles are talking about different things. Oleg Alexandrov (talk) 03:56, 10 March 2006 (UTC)

Power series example

About this example:

In an algebra of convergent power series at the origin, units are precisely those who do not vanish at the origin. The non-units (i.e., those vanish at the origin) then form a unique maximal idea, (thus, this algebra is a local ring.)

Is this mathematically incorrect? (I admit maybe I'm not understanding terminology correct, but I don't think the example itself is wrong.) -- Taku (talk) 21:18, 29 May 2008 (UTC)

It doesn't seem that interesting an example. The formal power series ring over any ring has the property that the units are exactly those with a unit constanrt term. "Convergent" just adds confusion. — Arthur Rubin (talk) 21:54, 29 May 2008 (UTC)

Oh, I thought "convergent" simplifies the idea. In my view, a formal power series is a generalization of a convergent power series. I thought the example was interesting because this is exactly how I learned about units and non-units. That's why I was surprised to find that the article didn't mean about this at all. -- Taku (talk) 06:25, 30 May 2008 (UTC)

Convergent where? On an interval containing 0? On a disk in the complex plane containing 0? (Well, those are the same, and it's well-defined.) I guess we differ on the appropriate level of abstraction. "Formal power series", or "convergent power series" = "analytic functions defined in a neighborhood of (or disk containing) 0, or the subring of rational functions defined at 0. They are are all local rings, but it's easier to see in the case of formal power series or rational functions than for convergent power series or analytic functions. — Arthur Rubin (talk) 13:56, 30 May 2008 (UTC)

By saying "at 0" I thought that would mean "in some neighborhood of 0" (so disk or interval, etc.) Maybe "around 0" or "near 0" is better wording? (I admit this might be a little bit imprecise.) Also, I didn't think formal power series or rational functions are easier to hand, because to see if a convergent power series is a unit or not, you only have to check it's inverse is analytic at 0 or not. Also, some adding some concrete familiar example is helpful for readers. Theoretically, maybe formal power series are easier to handle, but surely more readers are familiar with convergent power series. If the imprecision of the definition is what bothers you than that's easy to fix, I think. -- Taku (talk) 21:25, 30 May 2008 (UTC)

Perhaps just analytic functions rather than convergent power series. I mean, we know they're the same, but the fact that the reciprocal of an analytic function, non-zero at the specified point, is also analytic, is probably closer to "common sense" then that the inverse of a convergent power series is convergent. — Arthur Rubin (talk) 21:39, 30 May 2008 (UTC)

I didn't use the term "analytic functions" because then the example would be too simple to be interesting. (I also admit since the article now at least mentions about a local ring, this example ceases to be interesting, because you can say this algebra is just a local ring.) Anyway, the point is I didn't think it was confusing, but rather very prototypical example. If you disagree on this, then I'm fine with the removal of the example. -- Taku (talk) 21:56, 30 May 2008 (UTC)

But convergent power series are the same as analytic functions. It's not obvious, but it's stated in the analytic function article. — Arthur Rubin (talk) 22:22, 30 May 2008 (UTC)

Locally, of course (or depending on your definition of analyticity. I usually start with holomorphic = analytic.). Anyway, we are getting off-track. I noticed Local ring basically discusses an algebra of convergent power series. (since they start with continuous functions, you have to consider germs, not functions, though.) Since a germ of a holomorphic function is a power series, I still think "the algebra of convergent power series around 0" is a simple and typical example. What makes it not a "typical" example? -- Taku (talk) 23:10, 30 May 2008 (UTC)

Misleading Statement

The introduction says:

Unfortunately, the term ''unit'' is also used to refer to the identity element 1_''R'' of the ring, in expressions like ''ring with a unit'' or ''[[unit ring]]'', and also e.g. ''[[identity matrix|'unit' matrix]]''. (For this reason, some authors call 1_'''R''' "unity", and say that ''R'' is a "ring with unity" rather than "ring with a unit". Note also that the term ''[[unit matrix]]'' more usually denotes a matrix with all diagonal elements equal to one, and all other elements equal to zero.)

But "a matrix with all diagonal elements equal to one, and all other elements equal to zero" is the very meaning that was intended earlier, isn't it? If so, suggest removing "more usually". I don't know enough about ring theory to be sure, though, so I won't make the change myself. --BlueGuy213 (talk) 00:02, 16 March 2009 (UTC)

You're right. That last sentence is redundant. I'm going to remove it. Rckrone (talk) 03:01, 2 September 2009 (UTC)

Notation for unit group

We could consider using R^× instead of U(R) as the preferred notation for the unit group. I think that this more compact notation, introduced by Weil, is more common in the current mathematical literature, especially in algebraic number theory, which is probably the field in which unit groups are used most extensively. Another example: It is much more common to see C^× instead of U(C) as a notation for the set of nonzero complex numbers. Ebony Jackson (talk) 18:35, 27 December 2020 (UTC)

Well, there is also the notation $\mathbb {G} _{m}(R)$, which is probably not common outside algebraic geometry, though. For a non-commutative ring, I am not too sure if the notation $R^{\times }$ is that common (but I don't know much about the non-commutative ring literature.) -- Taku (talk) 05:06, 3 January 2021 (UTC)

I would say that not even algebraic geometers use $\mathbb {G} _{m}(R)$ as a notation for the unit group. It is just a group that happens to be the unit group. Ebony Jackson (talk) 05:46, 3 January 2021 (UTC)

I am confused; the notation $\mathbb {G} _{m}$ refers to the multiplicative group and the unit group of a commutative ring R is the same thing as multiplicative group of R; i.e., so, for me, by definition, $\mathbb {G} _{m}(R)$ = the group of unit elements. Maybe you're thinking of a different definition? -- Taku (talk) 05:50, 3 January 2021 (UTC)

Sorry for not being clearer. It is certainly true that $\mathbb {G} _{m}(R)=R^{\times }$. But I think that using $\mathbb {G} _{m}(R)$ as a notation for the unit group would be like writing $\operatorname {Frac} (\mathbb {Z} )$ instead of $\mathbb {Q}$ every time one wanted to refer to the field of rational numbers. Nobody does that! Ebony Jackson (talk) 06:00, 3 January 2021 (UTC)

Ah, I see. I think it matters of the perspective (a choice of the notation can certainly reflect that): in some theoretical sense, I think it is important (and worth mentioned in some form) that, for a commutative ring, $\mathbb {G} _{m}$ gives the multiplicative group of the ring, while $\mathbb {G} _{a}$ the underlying abelian group, the additive group of the ring. But I think I get your point is that the notation $\mathbb {G} _{m}(\mathbb {C} )$ does look weird. -- Taku (talk) 05:43, 4 January 2021 (UTC) (theoretically important, since, for example, the deformation theory should be able to give a sense to a group functor in between $\mathbb {G} _{m}$ and $\mathbb {G} _{a}$.)

By the way, I am not sure using the term "group scheme" is a good idea. Obviously that's what I had in mind but the article can and should be written to avoid the reference to scheme theory (the readers with an appropriate background can see we are talking about group scheme). -- Taku (talk) 05:50, 4 January 2021 (UTC)

I think the notation $\mathbb {G} _{m}$ is used more commonly for the group scheme that represents the functor, not so much the functor itself, so I'd say that if $\mathbb {G} _{m}$ is going to be mentioned at all, it is clearer to say that $\mathbb {G} _{m}$ is the group scheme that represents the functor R --> U(R). With GL_1, it is not so much of an issue, since it is reasonable to take the point of view that GL_1(R) is defined to be the group of invertible 1 x 1 matrices. I would probably omit the mention of $\mathbb {G} _{a}$; that is straying from the topic of units. Ebony Jackson (talk) 06:56, 4 January 2021 (UTC)

That's what I wanted to mean when I say "the notation $\mathbb {G} _{m}(R)$ is probably not common outside algebraic geometry". I suppose the question is: what is the common notation for the functor R --> U(R)? I don't think that's U (right?). At least in algebraic geometry, $\mathbb {G} _{m}$ seems like a common notation for that functor. Also, the point of mentioning $\mathbb {G} _{m}$ isn't really have a discussion on a particular group scheme but the fact that a unit element corresponds to a ring homomorphism $\mathbb {Z} [t,t^{-1}]\to R$. This seems important; it contrasts to the fact that a ring homomorphism $\mathbb {Z} [t]\to R$ corresponds to an element of R (so mentioning $\mathbb {G} _{a}$ seems to make sense). Also, mentioning this representability fact should also help somehow put group ring construction somehow in context. -- Taku (talk) 09:08, 4 January 2021 (UTC)

I would refer to it simply as " the functor R ↦ R^× ". Ebony Jackson (talk) 17:16, 4 January 2021 (UTC)

I agree with this completely and support changing the notation in the article, the biggest reason being simply that U(R) is less common than R^×. Joel Brennan (talk) 11:18, 23 July 2021 (UTC)

"Invertible element"

I believe that the terms "unit" and "invertible element" are synonymous in ring theory ("unit" in the sense of this article of course, not the multiplicative identity element) – if so then we should mention in the introduction that units also go by this name.

(note: "invertible element" is unambiguous in the context of ring theory since every element of a ring is additivity invertible, so clearly "invertible" must be refering to multiplicative invertibility) Joel Brennan (talk) 11:23, 23 July 2021 (UTC)

F

What is F in the example with integers? There is no reference to. Madyno (talk) 12:37, 23 December 2021 (UTC)

Thank you for pointing this out. The F is a number field, and R is its ring of integers. I fixed this (and a bunch of other little things!) Ebony Jackson (talk) 18:37, 29 December 2021 (UTC)

"Invertible element" listed at Redirects for discussion

An editor has identified a potential problem with the redirect Invertible element and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 December 10 § Invertible element until a consensus is reached, and readers of this page are welcome to contribute to the discussion. Jay 💬 16:21, 10 December 2022 (UTC)