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?

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. 


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 nonassociative algebra without pointing this out explicitly to your readers. 

