Talk:Associative algebra

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Untitled[edit]

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


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)


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[edit]

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)Reply[reply]

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

Ross Street[edit]

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)Reply[reply]

Inconsistencies[edit]

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)Reply[reply]

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)Reply[reply]

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)Reply[reply]

broken link[edit]

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

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Definition[edit]

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)Reply[reply]

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)Reply[reply]