Algebra / unital associative algebra: better terminology? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T09:06:36Z http://mathoverflow.net/feeds/question/15107 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15107/algebra-unital-associative-algebra-better-terminology Algebra / unital associative algebra: better terminology? S1 2010-02-12T14:28:53Z 2010-02-12T16:50:12Z <p>In Bourbaki an <em>algebra</em> 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 <em>prealgebra</em> for the first case or another short word used in the literature?</p> http://mathoverflow.net/questions/15107/algebra-unital-associative-algebra-better-terminology/15108#15108 Answer by Mariano Suárez-Alvarez for Algebra / unital associative algebra: better terminology? Mariano Suárez-Alvarez 2010-02-12T14:31:43Z 2010-02-12T14:31:43Z <p>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.</p> http://mathoverflow.net/questions/15107/algebra-unital-associative-algebra-better-terminology/15128#15128 Answer by Ben Webster for Algebra / unital associative algebra: better terminology? Ben Webster 2010-02-12T16:07:45Z 2010-02-12T16:07:45Z <p>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. </p>