Talk:Algebra over a field

message: This page states that the only finite dimensional alternative algebras over the reals are R, C, the quaternions and the octonions.  This doesn't seem correct, since any matrix ring is a finite dimensional alternative algebra over the reals, or any real vector space with xy := 0 as multiplication.  I don't know how to correct this -- maybe any finite dimensional alternative division algebra? --Sven 217.9.27.15 00:36, 20 Apr 2005 (UTC)
replies: text: Beats me. Maybe the statement is "any irreducible finite dimensional alternative algebra that is not an associative algebra"? (since a matrix ring is associative, and a real vector space with xy=0 but xx!=0 seems to be completely reducible, to me).linas 01:38, 20 Apr 2005 (UTC)
replies: text: I see no one fixed this in 15 months. What was probably meant was the Bruck-Kleinfeld theorem specialized to the field of real numbers: the only finite dimensional real alternative division algebras are R, C, the quaternions and octonions. The original author might also have meant Hurwitz' theorem: the only finite dimensional real normed alternative algebras are R, C, the quaternions and octonions. But I'm guessing the Bruck-Kleinfeld result is what was meant. --Michael Kinyon 12:19, 31 July 2006 (UTC)
thread_title: Alternative algebra bug?

message: I have big doubts about the validity of the section "index free notation".
* first, say why is it called like this
* second, should "k-algebra" be corrected to "K-algebra" everywhere ? (simultaneous use of K and k is very unfortunate ; in addition the prefix "k-" often means something else not related to any element "k" or set "K")
* third, according to the introduction, we do not suppose all algebras to be associative, thus they are not rings in general
* fourth, they should even less be supposed to be unital, and I think that eta(1) must be a unit if not eta=o
So I strongly suggest to move that paragraph (and the following, based upon it) away from the main page (e.g. into the  Talk page) until these issues are clarified. — MFH:Talk 15:02, 2 June 2006 (UTC)
thread_title:  index free notation ??? 

message: In the section « unital zero algebra », i read « (λ+u) (μ+v) = λμ + (λv+μu) ». Am i the only one chocked by the fact that scalars are freely added to vectors ? The law + is suposed to be an operation between vectors. I think correction is needed.  — Preceding unsigned comment added by 109.23.11.115 (talk) 16:44, 13 March 2015 (UTC)
replies: text: It does make sense as it stands; we are taking the direct sum of scalars and vectors by construction.  The notation is possibly confusing, since it is not distinguishing between a scalar and an element of the direct sum for which the vector part is zero, for example. This could be made more rigorous if it creates confusion, at the cost of verbosity. —Quondum 17:07, 13 March 2015 (UTC)
thread_title: Unital zero algebra

message: The link « unital algebra » redirects to the article « Ring ». I disagree : an unital algebra is not necessarily associative, whereas it is the case for rings... — Preceding unsigned comment added by 109.23.11.115 (talk • contribs) 2015-03-13T16:50:48
:This does seem to be a problem, and I agree.  , you changed the redirect; care to comment? —Quondum 17:01, 13 March 2015 (UTC)
replies: text: In any case, previous redirect to Pseudo-ring was worse. I have edited this article (rm  template) and redirected   here for fixing the issue. D.Lazard (talk) 17:33, 13 March 2015 (UTC)
replies: text: This seems to be the best route to go. There were still links to that redirect in this article, so I've removed them. —Quondum 20:46, 13 March 2015 (UTC)
thread_title: link in section « unital algebra »

message: These algebras are just monoid objects in the category of vector spaces (with tensor product as monoidal product.) Also, algebra homomorphisms are just monoid object homeomorphisms. I am not sure where to put it though. TheKing44 (talk) 21:45, 12 May 2015 (UTC)
replies: text: I'm not arguing definitively against inserting it, but my take is that it's a fairly isolated statement that might be awkward to fit in. We could add such a line to numerous other articles. The line rightly belongs (and already appears) in the monoid (category theory) article. Rschwieb (talk) 12:48, 13 May 2015 (UTC)
thread_title:  Should we put that they are just monoid objects? 

message: Substituted at 01:44, 5 May 2016 (UTC)
thread_title: Assessment comment

message: At this moment the first sentence of this article reads: "In mathematics, an algebra is one of the fundamental algebraic structures used in abstract algebra."
This sounds quite circular and rather funny. I suggest the sentence ought to be modified to remove the ha ha factor. — Preceding unsigned comment added by Dratman (talk • contribs) 01:00, 11 August 2016 (UTC)
replies: text: I agree. I have edited the lead for fixing this and removing a WP:PEACOCK term ("fundamental"). D.Lazard (talk) 08:45, 11 August 2016 (UTC)
thread_title:  Is the article's first sentence intended to be humorous? 

message: They are incredibly helpful to understand "why the heck would anyone care about this construction", and sorely missing or buried later in the text in too many Wikipedia math articles. 50.197.48.113 (talk) 14:43, 2 February 2024 (UTC)
thread_title:  Thank you for the motivating examples