Talk:Unit (ring theory)
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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)
[edit] 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)
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- 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)
- 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)
- 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)
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)
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- 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)
