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Questions of the kind "What's the name for a X that satisfies property Y?"

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Algebra / unital associative algebra: better terminology?

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. …