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!

Note that modules only have one "internal" operation, addition; algebra has two such operations, addition and multiplication.
Closing something under more operations requires more. In particular if you require that $x^n\neq 1$ for all $n$ for the generators.
Observe $\mathbb{N,Z,Q}$. All are generated by the number $1$. Only that $\mathbb N$ requires only closure under addition; $\mathbb Z$ requires also closure under subtraction; and $\mathbb Q$ requires closure under division by non-zero elements.
Similarly if we take $x$ and we add it to a [unital commutative] ring $R$, if we only wish to define $x+x$ and so on, we get a module generated by $x$, that is $R+Rx$. If, on the other hand, we want to allow $x^n$ we effectively get all the polynomials in $x$ (granted we don't want $x$ to be a torsion element), namely $R[x]$.
