Talk:Associative algebra

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Content of Associative Algebra, now redirected here: Charles Matthews 17:24, 11 Oct 2004 (UTC)

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Sorry to have disturbed you, Charles, by adding a page on 'Associative_Algebra'. I moved the text to Semigroups. Nico Benschop 11:19, 12 Oct 2004 (UTC)

Well actually, semigroup exists. May I suggest the list of abstract algebra topics?

Charles Matthews 11:38, 12 Oct 2004 (UTC)

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It is not uncommon for an algebra to be defined over a commutative ring (with 1). There is nothing special about using a field. In fact this view should be encouraged. It illustrates the special role that the integers (Z) play in the polynomial ring Z[x] for instance.

Serious Errors

There are some serious errors with this article, especially the parts on coalgebras, Hopf algebras, and Lie algebras. It seems to confuse tensor products with Cartesian products and Lie algebras with their enveloping algebras. Also the product and coproduct in a bialgebra are NOT unrelated, the coproduct is an algebra homomorphism and the product is a coalgebra homomorphism. Moreover there are already much higher quality articles on these topics. Fiedorow 21:22, 16 December 2005 (UTC)

Please supply what corrections you can. Charles Matthews 21:51, 16 December 2005 (UTC)

My feeling is that this shouldn't be fixed by just editing this article. I think some of the discussion should be moved elsewhere. Perhaps a new article on linear representations of algebras. However that should probably include a discussion of infinite dimensional representations, which I am not qualified to edit. Fiedorow 01:29, 17 December 2005 (UTC)

Ross Street

Reference section lacks ISBN, format, etc. - and the book isn't in Cornell's library nor in the Library of Congress though I shall look elsewhere. Amazon.com notes a book by Prof? Street entitled

Quantum Groups: A Path to Current Algebra (Australian Mathematical Society Lecture Series) (Paperback) (ISBN 0521695244)

to be released sometime around December this year (or after?) by Cambridge University Press. Don't know if that will contain the same material. Alternative references (well- maybe not after all for instance Richard S. Pierce's "Associative Algebras", Springer-Verlag, ISBN 0-387-90693-2, 1982 for the copy I have that's in front of me ... but unlike the one listed, this one should be available in a library. (Don't know how many do, WorldCat/RLIN check where possible is worth doing from our site for certain kinds of references, I think!) Schissel | Sound the Note! 13:44, 5 September 2006 (UTC)

Inconsistencies

As of Dec. 30 2009, there are some inconsistencies in this article. The main definition defines an associative but not necessarily unital algebra over a commutative ring R. But the constructions "from R-modules" and "from rings" only give ways to build every possible associative unital algebra over a commutative ring.

I find nonunital algebras quite annoying, but if that's what is under discussion in this page, the construction "from R-modules" should leave out the identity 1, and the construction "from rings" should allow A to be "rng" --- a ring without unit --- and it should only demand that the homomorphisms from R to A be a rng homomorphism. The former correction is pretty painless. The latter correction is more annoying, since most people don't know about rngs. So perhaps one might just keep the current construction "from rings" but admit that one is only getting associative unital algebras this way.

Someone should also come out and say that for lots of people, "associative algebra" means "associative unital algebra". Indeed, associative but not necessarily unital algebras should be just as unpopular (or popular) as rngs. John Baez (talk) 21:22, 30 December 2009 (UTC)

Worse, the definition here of "algebra homomorphism" assumes our algebra has a unit, while the definition of "algebra" does not! Aargh! John Baez (talk) 21:25, 30 December 2009 (UTC)

I'm going to temporarily take matter into my own hands and fix things as I see fit. More carefully considered fixes are welcome. Both unital and not-necessarily-unital associative algebras deserve their place in the sun, but we need a consistent notation. John Baez (talk) 21:26, 30 December 2009 (UTC)

broken link

The link to Street's paper is broken. Crasshopper (talk) 01:07, 8 August 2013 (UTC)

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Definition

part of the definition says "where the ring addition and module addition are the same", I find the "are the same" unclear so maybe the property they satisfy should be made explicit — Preceding unsigned comment added by 2001:4CA0:0:F234:8DB7:F4F7:B0FD:15FA (talk) 14:36, 18 October 2021 (UTC)

It is not "are the same", it is "are the same operation". Nevertheless, I have edited the sentence for improving the clarity. D.Lazard (talk) 16:14, 18 October 2021 (UTC)

Merge with Algebra homomorphism

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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I propose that we merge algebra homomorphism into this article. The reason is that I am not convinced that such a separate article is needed. Note we already have ring homomorphism, which mostly covers what such an article can cover. There are few aspects specific to algebra homomorphisms but that can be easily handled in this article. Taku (talk) 10:04, 3 March 2023 (UTC)

• Oppose. Instead, Algebra homomorphism must be fixed for including all sorts of algebra homomorphisms. Presently, for Wikipedia, a Lie algebra is an algebra, but a Lie algebra homomorphism is not an algebra homomorphism. This is ridiculous. Also, the section Algebra over a field#Algebra homomorphisms has a template {{main|Algebra homomorphism}}, while the section includes non-associative algebras, and the main article, does not include them presently (I’ll edit the first sentence of Algebra homomorphism for fixing this). The merge would be make things worse. D.Lazard (talk) 11:10, 3 March 2023 (UTC)

  I should have emphasized that my proposal is based on the current state of the article (as you can see currently it doesn't have much stuff that is not already in this article). If it were to be rewritten to cover the non-associative case, then obviously the merger doesn't make sense. But I do wonder if we need an algebra homomorphism article for a very general algebraic structure. Isn't it just a structure-preserving map? -- Taku (talk) 11:23, 3 March 2023 (UTC)

  The problem is that before my edit the content of the article did not correspond to the standard meaning of the title. The proposed merge would suggest to non-experts that algebra homomorphisms are restricted to associative algebras, and that a Lie algebra homomorphism is not an algebra homomorphism. I would not oppose to merge algebra homomorphism into a dedicated section of homomorphism. This would make sense, as this article has no real content, except the definition and a few examples. D.Lazard (talk) 13:56, 3 March 2023 (UTC)

  It's really a matter of context. An algebra can be associative or not or unital or not and that depends on context. We cannot quite argue that an algebra should mean a possibly non-associative one (much like, "going to American" could mean going to the US or a non-US place and we cannot dictate on people's usage.) Anyway, yes, maybe the homomorphism article is a better place for discussing homomorphisms between general algebraic structures. Like said initially, I am not convinced there are much materials on algebra homomorphisms in the general setup, that cannot be discussed in the associative algebra article. -- Taku (talk) 07:59, 4 March 2023 (UTC)

• Support I'm thinking every different article on each kind of algebra can have a subsection for homomorphisms for that particular kind of algebra. That way, specifics, exceptions, oddities for that kind of algebra can be covered in detail. It is hard for me to imagine that most readers will want a single article that compares & contrasts all the different kinds of algebra homomorphisms that might arise. 67.198.37.16 (talk) 23:47, 13 June 2023 (UTC)
• Comment I still oppose to the proposed merge. However I would support to merge Algebra homomorphism into Algebra over a field. This would be coherent with the modern use that every structure goes with a corresponding concept of homomorphism. Such a merge would not cover the case of algebras over a ring, but this problem has no satisfactory solution, except merging Algebra over a ring into Algebra over a field and moving the result to Algebra (structure). D.Lazard (talk) 07:47, 14 June 2023 (UTC)

  Comment It seems that particular merge already happened, back in 2018, just without changing the page title. Felix QW (talk) 20:06, 18 November 2023 (UTC)

• Support merging this material somewhere rather than duplicating it on its own page. This article consists of a definition with some simple consequences and a couple of subnotions. It then goes on to give some examples. My suggestion would be to integrate the content where it is useful, whether that be at associative algebra, non-associative algebra or algebra over a field, possibly more than one or none at all, and then redirect the title somewhere. There is one other conflicting usage I know of, which is that of a homomorphism of algebras in universal algebra. Perhaps these ambiguities would even justify making a DAB page for algebra homomorphism? Felix QW (talk) 20:17, 18 November 2023 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

I know this is closed discussion, but it does seem to me to be a perverse closure result, given the meaning of the term and the contested nature. FWIW, I fully support D.Lazard's position on this. Even if this were a stub, and merging to being a section of an encapsulating article were indicated as per the support for merge above, merging to the chosen target (Associative algebra) is actively putting misinformation/confusion into the article at a structural level. The unwary novice may be induced to think that algebras are, by definition, associative. We should avoid any such implication. The proposed merge, as is, would also imply either removing information about non-associative algebras, or include non-associative algebras in a section as a generalization in the of associative algebras, both of which are a bit perverse (especially as Non-associative algebra exists with no mention of homomorphisms). —Quondum 13:19, 5 April 2024 (UTC)

Although it was a clear consensu to merge or redirect Algebra homomorphism somewhere, there was no consensus on the target. Also, the closing comment is "merge" without indication on the target. So, I have redirected boldy Algebra homomorphism to Algebra over a field#Algebra homomorphisms. D.Lazard (talk) 14:15, 5 April 2024 (UTC)

Ah, okay. Perfect. I came across this because of the banner at the top of Algebra homomorphism that you have now taken care of. Thanks. —Quondum 17:35, 5 April 2024 (UTC)

Semimartingales

In the examples, the article mentions that semimartingales form a ring. But, in this article, we're talking about associative algebras, not rings. Do the semimartingales also form an associative algebra? Thatsme314 (talk) 05:52, 9 October 2023 (UTC)

Terrible

What a terrible explaination 2A02:14F:178:3BA5:0:0:C665:9328 (talk) 22:44, 22 November 2023 (UTC)