can anyone help me with the following statement (it is part of a bigger proof where it is not explained).

Let $B$ be a finite type commutative $A$-algebra (where $A$ is a commutative ring), and consider the kernel $I$ of the diagonal homomorphism $B\otimes_A B\to B$ (defined by $b\otimes b'\mapsto bb'$). Then $I$ is a finitely generated ideal.

My guess is the following: if $b_1,\ldots,b_n$ generate $B$ over $A$ (as an algebra), then the elements $b_i\otimes 1 - 1\otimes b_i$ are the desired generators of $I$. Is there an easy way to see this?

Thanks in advance.