# Algebra / unital associative algebra: better terminology?

In Bourbaki an algebra over a commutative ring $k$ is defined to be a $k$-module $A$ together with a $k$-bilinear map $A \times A \rightarrow A$. We then have the obvious notion of morphisms of $k$-algebras. This terminology is nice, because e.g. Lie algebras are then a special kind of algebras and so on. But in 95% of my work I am using unital associative $k$-algebras with unital morphisms. Now, is there any better (in particular shorter) terminology available to distinguish these two cases? I don't want to add this "unital associative" and "unital morphisms" all the time. Is perhaps something like prealgebra for the first case or another short word used in the literature?

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Very tiny comment: the spelling is unital, not unitial. I wouldn't have dreamed of mentioning it except that the question is about how to use words. –  Tom Leinster Feb 12 '10 at 16:40
Wow, that was a (repeated!) typo. Thanks. –  user717 Feb 12 '10 at 16:50
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## 2 Answers

If your work requires you to work mostly with unital assosiative $k$-algebras with unital morphisms, define somewhere prominent in your work 'algebra' and 'morphism of algebras' to mean precisely that, and say 'Lie algebras', 'not necessarily associative algebra', 'possibly non unital morphism of algebra', and so on in the meagre 5% of the remaining cases.

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After thinking about this for a day, I'll do what you suggested :) –  user717 Feb 13 '10 at 14:41
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I will mention: most mathematicians of my acquaintance do not agree with Bourbaki's notation. As far as I'm concerned, an algebra is unital associative. In fact, I would discourage you from using the word "algebra" to mean a possibly non-associative algebra without pointing this out explicitly to your readers.

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Agreed. I have even seen the convention that "algebra" = "unital associative algebra" and "associative algebra" = "associative, not necessarily unital algebra". This is an instance of the (Bourbakiste!) idea that the most used concept should have the simplest name. –  Pete L. Clark Feb 12 '10 at 16:18
Yes, I agree . –  user717 Feb 12 '10 at 16:20
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