What's the difference between finite and finitely generated algebras

Question

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

Answer 1

$B[a_1, a_2, ... , a_n]$ means the ring of polynomials of $a_1, a_2, ... , a_n$ with coefficients in $B$ i.e. $$B[a_1, a_2, ... , a_n] = \{f(a_1, a_2, ..., a_n) : f \in B[x_1, x_2, ..., x_n]\}$$ where $x_1, ..., x_n$ are distinct variables. It is not always finitely generated $B$-module what is actually said in $(2)$.

Answer 2

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.