Talk:Algebra over a field

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It's nice to see this separated out into its own thing as it should be. The problem is that there's really no good name for it. "algebra over a field" isn't quite right, since algebras can be defined over any commutative ring with no change in the concept, and one does study these things with the same terminology and notation in (for example) algebraic geometry. But "algebra over a commutative ring" isn't too hot either, being less familiar and (with "commutative") rather unwieldy.

One idea is "algebra over a base", since the commutative ring in question is called the base ring (or base field). One does hear phrases such as "Let A be an algebra over the base ring K.", for example. But I can easily imagine that this means something else too. Another idea is "algebra over a base ring", which has some of the problems of the version with "commutative" above, but is not so enormous.

Ideas?

-- Toby 06:39 Mar 7, 2003 (UTC)

Well, while editing this page, I decided that it's OK as it is. The fact the "vector space" and "module" are completely different terms is a much bigger problem (and much bigger than Wikipedia) that already prevents discussing the general case in the same breath as the more familiar. So let it be. -- Toby 07:46 Mar 7, 2003 (UTC)

[edit] Alternative algebra bug?

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)

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)

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)

[edit] index free notation ???

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)