Why Does Finitely Generated Mean A Different Thing For Algebras? I've always wondered why finitely generated modules are of form 
$$M=Ra_1+\dots+Ra_n$$
while finitely generated algebras have form 
$$R=k[a_1,\dots, a_n]$$
and finite algebras have form 
$$R=ka_1+\dots +ka_n$$
It seems to me that this is an flagrant abuse of nomenclature, and has certainly confused me in the past. Is there some historical reason for this terminology? Any references to the genesis of these terms would be greatly appreciated!

A module is an additive structure. Module elements can be added together, not multiplied. So a finitely-generated module should consist of linear combinations of finitely many module elements. 
An algebra has both multiplication and addition. The notation for a finitely generated module doesn't communicate that "generators" can be multiplied together. Now there are finitely many algebra elements that are combined in the legal ways (using addition, subtraction, and multiplication by other generators). The new notation $k[a_1,\ldots,a_n]$ tries to communicate this, if only be being something different thatn the module notation.
To me, "finite algebra" is trying to communicate that the algebra is finite dimensional. And we are back to having an additive concept (dimension). So the notation is back to the module style.
