Simple question about finitely generated algebras

I have two finitely generated algebras $A$ and $B$ over a field $\mathbb{K}$ such that $B\subseteq A$. Is it true that $A=B[a_1,\ldots,a_n]$ for some $a_1,\ldots,a_n\in A$?

Motivation: I am trying to prove that for finitely generated algebras the notions of integral extension and finite extension coincide. I managed to prove that extension $B\subseteq A$ is finite iff $A=B[a_1,\ldots,a_n]$ for some elements $a_1,\ldots,a_n\in A$ integral over $B$. Thus, if the answer to my question is positive, everything is proved.

Solution 1:

If $A$ is finitely generated over $K$ then we have $A=K[a_1,...,a_n]$ for some elements $a_1,...,a_n\in A$. Then:

$A=K[a_1,...,a_n]\subseteq B[a_1,...,a_n]\subseteq A$