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associator

Defines: 
anti-associative
Type of Math Object: 
Definition
Major Section: 
Reference
Groups audience: 

Mathematics Subject Classification

17A01 no label found

Comments

couldn't the associater be defined for rings? They seem to have all necessary operations.

HkBSt, the proper way to fix this is not by posting underneath the entry, but filling a correction (the link for that is below the entry when you view it)

the uathor then gets the notice and proceeds to change what's required. So if you find most of these mistakes , please point them to the author using corrections system (besides, that way author gets an email akerting them, in the way you did, will only be fixed if the author looks at the entry)
f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

Sure, the associator can be defined for rings. But it is rather useless, since a ring is, by definition, multiplicatively associative and hence, has a trivial associator. When we're speaking of not-necessarily associative algebras, we are talking about a "generalized" ring (over a field) in which the multiplication is not assumed to be associative.

But logically it should be defined for something which I will call here a nonassociative ring. The extra operation of scalar multiplication that an algebra has over a ring is superfluous.

Hmmm, it also seems there is no definition of a generalized ring on PM. Maybe I will make an entry sorting this stuff out.

You should be careful when using the phrase "nonassociative ring" since it implies the exclusion of any "associative" ring, unless this is exactly your intent. I prefer the naming of a "generalized ring" to include both the traditional "associative" rings as well as the nonassociative ones. I am not aware of any other naming convention for a "generalized" ring in the mathematical community.

But to avoid any more confusion, please add an entry and let's go from there.

Chi

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