TITLE: What's the difference between finite and finitely generated algebras
QUESTION [7 upvotes]: I didn't understand the difference between the two definitions:

I thought the definition of $B[a_1,\ldots,a_n]$ is exactly the one in the item (b), i.e., $B[a_1,\ldots,a_n]=Ba_1+\ldots+Ba_n$. 
I need help
Thanks

REPLY [10 votes]: An example: the polynomial $k[X]$ is a finitely generated $k$-algebra bit not a finite $k$-algebra.
An alternative, and possibly more illuminating, form of those definitions is the following: a ring $B$ contain a subring $A$ is

finitely generated over $A$ if there is a finite subset $S\subseteq B$ such that the smallest subring of $B$ containing $A$ and $S$ is $B$ itself, and
finite over $A$ if there is a finite subset $S\subseteq B$ such that the smallest $A$-submodule of $B$ containing $A$ and $S$ is $B$ itself.